Research Article Wave-Breaking Criterion for the ...
11
Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2013, Article ID 809824, 10 pages http://dx.doi.org/10.1155/2013/809824 Research Article Wave-Breaking Criterion for the Generalized Weakly Dissipative Periodic Two-Component Hunter-Saxton System Jianmei Zhang and Lixin Tian Department of Mathematics, Nonlinear Scientific Research Center, Jiangsu University, Zhenjiang, Jiangsu 212013, China Correspondence should be addressed to Jianmei Zhang; [email protected] Received 22 May 2013; Accepted 22 July 2013 Academic Editor: Michael Meylan Copyright © 2013 J. Zhang and L. Tian. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper studies the wave-breaking criterion for the generalized weakly dissipative two-component Hunter-Saxton system in the periodic setting. We get local well-posedness for the generalized weakly dissipative two-component Hunter-Saxton system. We study a wave-breaking criterion for solutions and results of wave-breaking solutions with certain initial profiles. 1. Introduction In recent years, the Hunter-Saxton equation [1] + 2 + =0 (1) models the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal. In Hunter and Saxton [1], is the space variable in a reference frame moving with the linearized wave velocity, is a slow-time variable, and (, ) is a measure of the average orientation of the medium locally around at time . In order to be more precise, the orientation of the molecules is described by the field of unit vectors (cos (, ), sin (, )) [2]. e Hunter- Saxton equation also arises in a different physical context as the high-frequency limit [3, 4] of the Camassa-Holm equation for shallow water waves [5, 6] and a reexpression of the geodesic flow on the diffeomorphism group of the circle [7] with a bi-Hamiltonian structure [1, 8] which is completely integrable [4, 9]. Hunter and Saxton [1] explored the initial value problem for the Hunter and Saxton equation on the line (nonperiodic case) and on the unit circle = / by using the method of characteristics, while Yin [2] studied it by using the Kato semigroup method. In addition, the two classes of admissible weak solutions, dissipative and conservative solutions, and their stability were investigated in [10–12]. Lenells [13] confirmed that the Hunter-Saxton equation also describes the geodesic flows on the quotient space of the infinite-dimensional group () modulo the subgroup of rotations Rot(). e Camassa-Holm equation admits many integrable multicomponent generalizations. So many authors studied the two-component Camassa-Holm system [14, 15]. Inspired by this, recently, the researchers have made a study of the global existence of solutions to a two-component generalized Hunter-Saxton system in the periodic setting as follows: + 2 + − + = 0, > 0, ∈ , + () = 0, > 0, ∈ , (, + 1) = (, ) , (, + 1) = (, ) , ≥ 0, ∈ , (0, ) = 0 () , (0, ) = 0 () , ∈ . (2) e authors of [16] have explored the particular choice of the parameter =1. e authors of [17] have further studied the wave breaking and global existence for the system for the parameter ∈ R to determine a wave-breaking criterion for strong solutions by using the localization analysis in the transport equation theory. In general, avoiding energy dissipation mechanisms in a real world is not so easy. Wu and Yin [18, 19] have investigated the blow-up phenomena and the blow-up rate of the strong
Transcript of Research Article Wave-Breaking Criterion for the ...
Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 809824 10 pageshttpdxdoiorg1011552013809824
Research ArticleWave-Breaking Criterion for the Generalized Weakly DissipativePeriodic Two-Component Hunter-Saxton System
Jianmei Zhang and Lixin Tian
Department of Mathematics Nonlinear Scientific Research Center Jiangsu University Zhenjiang Jiangsu 212013 China
Correspondence should be addressed to Jianmei Zhang jianmeizhangsinacom
Received 22 May 2013 Accepted 22 July 2013
Academic Editor Michael Meylan
Copyright copy 2013 J Zhang and L TianThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper studies the wave-breaking criterion for the generalized weakly dissipative two-component Hunter-Saxton system in theperiodic setting We get local well-posedness for the generalized weakly dissipative two-component Hunter-Saxton system Westudy a wave-breaking criterion for solutions and results of wave-breaking solutions with certain initial profiles
1 Introduction
In recent years the Hunter-Saxton equation [1]
119906119905119909119909
+ 2119906119909119906119909119909+ 119906119906119909119909119909
= 0 (1)
models the propagation of weakly nonlinear orientationwaves in a massive nematic liquid crystal In Hunter andSaxton [1] 119909 is the space variable in a reference framemovingwith the linearized wave velocity 119905 is a slow-time variableand 119906(119905 119909) is a measure of the average orientation of themedium locally around 119909 at time 119905 In order to be moreprecise the orientation of the molecules is described by thefield of unit vectors (cos 119906(119905 119909) sin 119906(119905 119909)) [2] The Hunter-Saxton equation also arises in a different physical contextas the high-frequency limit [3 4] of the Camassa-Holmequation for shallow water waves [5 6] and a reexpressionof the geodesic flow on the diffeomorphism group of thecircle [7] with a bi-Hamiltonian structure [1 8] which iscompletely integrable [4 9] Hunter and Saxton [1] exploredthe initial value problem for the Hunter and Saxton equationon the line (nonperiodic case) and on the unit circle 119878 =
119877119885 by using the method of characteristics while Yin [2]studied it by using the Kato semigroup method In additionthe two classes of admissible weak solutions dissipative andconservative solutions and their stability were investigatedin [10ndash12] Lenells [13] confirmed that the Hunter-Saxtonequation also describes the geodesic flows on the quotient
space of the infinite-dimensional group 119863119904(119878) modulo the
subgroup of rotations Rot(119878)The Camassa-Holm equation admits many integrable
multicomponent generalizations So many authors studiedthe two-component Camassa-Holm system [14 15] Inspiredby this recently the researchers have made a study of theglobal existence of solutions to a two-component generalizedHunter-Saxton system in the periodic setting as follows
119906119905119909119909
+ 2120590119906119909119906119909119909+ 120590119906119906
119909119909119909minus 120588120588119909+ 119860119906119909= 0
119905 gt 0 119909 isin 119877
120588119905+ (120588119906)
119909= 0 119905 gt 0 119909 isin 119877
119906 (119905 119909 + 1) = 119906 (119905 119909) 120588 (119905 119909 + 1) = 120588 (119905 119909)
119905 ge 0 119909 isin 119877
119906 (0 119909) = 1199060(119909) 120588 (0 119909) = 120588
0(119909) 119909 isin 119877
(2)
The authors of [16] have explored the particular choice ofthe parameter 120590 = 1 The authors of [17] have further studiedthe wave breaking and global existence for the system for theparameter 120590 isin R to determine a wave-breaking criterionfor strong solutions by using the localization analysis in thetransport equation theory
In general avoiding energy dissipation mechanisms in areal world is not so easyWu and Yin [18 19] have investigatedthe blow-up phenomena and the blow-up rate of the strong
2 Journal of Applied Mathematics
solutions of the weakly dissipative CH equation and DPequation Inspired by the results mentioned above we aregoing to discuss the initial value problem associated withthe generalized weakly dissipative periodic two-componentHunter-Saxton system
119906119905119909119909
+ 2120590119906119909119906119909119909+ 120590119906119906
119909119909119909minus 120588120588119909+ 119860119906119909
+ 120582 (119906 minus 119906119909119909) = 0 119905 gt 0 119909 isin 119877
120588119905+ (120588119906)
119909= 0 119905 gt 0 119909 isin 119877
119906 (119905 119909 + 1) = 119906 (119905 119909) 120588 (119905 119909 + 1) = 120588 (119905 119909)
119905 ge 0 119909 isin 119877
119906 (0 119909) = 1199060(119909) 120588 (0 119909) = 120588
0(119909) 119909 isin 119877
(3)
where 120590 isin 119877 is the new free parameter and 119860 ge 0 120582 lt 0Our major results of this paper are Theorems 11 and
12 (wave-breaking criterion) The remainder of the paperis organized as follows Section 2 establishes the local well-posedness for (3) with the initial data in 119867
119904times 119867119904minus1 119904 ge 2
Section 3 deals with the wave breaking of this new systemTheorem 11 using transport equation theory states a wave-breaking criterion which says that the wave breaking onlydepends on the slope of 119906 not the slope of 120588 Theorem 12improves the blow-up criterion with a more precise condi-tion
Notation Throughout this paper 119878 = 119877119885 will denote theunit circle By119867119904 119904 ge 0 we will represent the Sobolev spacesof equivalence classes of functions defined on the unit circle119878 which have square-integrable distributional derivatives upto order 119904 The119867119904-norm will be designated by sdot
119867119904 and the
norm of a vector V isin 119867119904 times 119867119904minus1 will be written as V119867119904times119867119904minus1
Also the Lebesgue spaces of order 119901 isin [1infin]will be denotedby 119871119901(119878) and the norm of their elements will be denoted
by 119891119871119901(119878) Finally if 119901 = 2 we agree on the convention
sdot 1198712(119878)
= sdot
2 Preliminaries
In this part we will establish the local well-posedness forthe Cauchy problem of system (3) by using Katorsquos theory Topursue our goal we give the results we wanted in brief
We now provide the framework in which we will refor-mulate (3) To do this we observe that we can write the firstequation of (3) in the following integrated form
119906119905119909+
120590
2
1199062
119909+ 120590119906119906
119909119909minus
1
2
1205882+ 119860119906 + 120582120597
minus1
119909119906 minus 120582119906
119909= 119892 (119905)
(4)
where 120597minus1119909119891(119909) = int
119909
0119891(119910)119889119910 and 119892(119905) is determined by the
periodicity of 119906 to be
119892 (119905) = minusint
119878
(
120590
2
1199062
119909+
1
2
1205882minus 119860119906)119889119909 (5)
Integrating both sides of (4) with respect to variable 119909we get
119906119905+ 120590119906119906
119909= 120597minus1
119909(
120590
2
1199062
119909+
1
2
1205882minus 119860119906 + 119892 + 120582119906
119909minus 120582120597minus1
119909119906)
+ ℎ (119905)
(6)
where ℎ(119905) [0infin) rarr 119877 is an arbitrary continuous functionTherefore (3) can be written in the ldquotransportrdquo form asfollows
119906119905+ 120590119906119906
119909= 120597minus1
119909(
120590
2
1199062
119909+
1
2
1205882minus 119860119906 minus 120582120597
minus1
119909119906 + 120582119906
119909+ 119892)
+ ℎ (119905) 119905 gt 0 119909 isin 119877
120588119905+ 119906120588119909= minus119906119909120588 119905 gt 0 119909 isin 119877
119906 (119905 119909 + 1) = 119906 (119905 119909) 120588 (119905 119909 + 1) = 120588 (119905 119909)
119905 ge 0 119909 isin 119877
119906 (0 119909) = 1199060(119909) 120588 (0 119909) = 120588
0(119909) 119909 isin 119877
(7)
where ℎ(119905) [0infin) rarr 119877 is an arbitrary continuous functionNext we apply Katorsquos theory to establish the local well-
posedness for the system (3) Consider the abstract quasi-linear evolution equation
119889V
119889119905
+ 119860 (V) V = 119891 (V) 119905 ge 0 V (0) = V0 (8)
Proposition 1 (see [20]) Given the evolution equation (8)assume that the Kato conditions hold For a fixed V
0isin 119884 there
is a maximal 119879 gt 0 depending only on V0119884and a unique
solution V to the abstract quasi-linear evolution equation (8)such that
V = V (sdot V0) isin 119862 ([0 119879) 119884) cap 119862
1([0 119879) 119883) (9)
Moreover the map V0rarr V(sdot V
0) is continuous from 119884 to
119862 ([0 119879) 119884) cap 1198621([0 119879) 119883) (10)
One may follow the similar argument as in [17] to obtain thefollowing local well-posedness for (3)
Theorem 2 Given any 1198830= (1199060
1205880
) isin 119867119904times 119867119904minus1 119904 ge 2 there
exist a maximal 119879 = 119879(120590 119860 1198830119867119904times119867119904minus1) gt 0 and a unique
solution119883 = (119906
120588 ) to (3) such that
119883 = 119883 (sdot 1198830) isin 119862 ([0 119879) 119867
119904(119878) times 119867
sminus1(119878))
cap 1198621([0 119879) 119867
119904minus1(119878) times 119867
119904minus2(119878))
(11)
Moreover the solution depends continuously on the initialdata that is the mapping 119883
0rarr 119883(sdot 119883
0) 119867
119904times 119867119904minus1
rarr
119862([0 119879)119867119904(119878) times 119867
119904minus1(119878)) cap 119862
1([0 119879) and 119867
119904minus1(119878) times
119867119904minus2(119878)) is continuous and the maximal existence time 119879 can
be chosen independently of the Sobolev order 119904
Journal of Applied Mathematics 3
Now discuss the initial value problem for the Lagrangian flowmap as follows
120597120593
120597119905
= 119906 (119905 120593 (119905 119909)) 119905 isin [0 119879)
120593 (0 119909) = 119909 119909 isin 119877
(12)
where 119906 is the first component of the solution 119883 to (3) Usingclassical results from ordinary differential equations one canacquire the following result on 120593which is of vital importance inthe proof of the blow-up scenarios
Lemma 3 (see [17]) Let 119906 isin 119862([0 119879)119867119904) cap 1198621([0 119879)119867119904minus1)119904 ge 2 Then initial value problem (12) admits a unique solu-tion 120593 isin 119862
1([0 119879) times 119877 119877) Moreover 120593(119905 sdot)
119905isin[0119879)is increas-
ing diffeomorphism of 119877 with
120593119909(119905 119909) = 119890
int119879
0
119906119909(120591120593(120591119909))119889120591
gt 0 (119905 119909) isin [0 119879) times 119877 (13)
Remark 4 Since 120593(119905 sdot) 119877 rarr 119877 is a diffeomorphism ofthe linear for every 119905 isin [0 119879) the 119871infin-norm of any functionV(119905 sdot) isin 119871
infin 119905 isin [0 119879) is preserved under the family ofdiffeomorphisms 120593(119905 sdot) with 119905 isin [0 119879) that is
V (119905 sdot)119871infin(119878)
=1003817100381710038171003817V (119905 120593 (119905 sdot))100381710038171003817
1003817119871infin(119878) 119905 isin [0 119879) (14)
Similarly we have
inf119909isin119878
V (119905 119909) = inf119909isin119878
V (119905 120593 (119905 119909)) 119905 isin [0 119879)
sup119909isin119878
V (119905 119909) = sup119909isin119878
V (119905 120593 (119905 119909)) 119905 isin [0 119879)
(15)
Lemma 5 Let 1198830= (1199060
1205880
) isin 119867119904times 119867119904minus1 119904 ge 2 and let 119879 be
the maximal existence time of the solution119883 = (119906
120588 ) to (3) withinitial data 119883
0 Then for all 119905 isin [0 119879) we have the following
results
int
119878
120588 (119905 119909) 119889119909 = int
119878
1205880(119905) 119889119909 (16)
int
119878
1199062
119909(119905 119909) + 120588
2(119905 119909) 119889119909 le int
119878
1199062
0119909(119909) + 120588
2
0(119909) 119889119909 ≜ 119864
0
(17)
Proof On the one hand integrating the second equation in(3) by parts and using the periodicity of 119906 and 120588 we acquire
119889
119889119905
int
119878
120588 119889119909 = minusint
119878
(119906120588)119909119889119909 = 0 (18)
On the other hand multiplying (4) by 119906119909and integrating
by parts considering the periodicity of 119906 we obtain
119889
119889119905
int
119878
1199062
119909119889119909 = minus2int
119878
119906120588120588119909119889119909 + 2120582int
119878
1199062119889119909 + 2120582int
119878
1199062
119909119889119909
(19)
Multiplying the second equation in (3) by 120588 and integrat-ing by parts we have
119889
119889119905
int
119878
1205882119889119909 = 2int
119878
119906120588120588119909119889119909 (20)
Adding the above two equations we get
119889
119889119905
int
119878
1199062
119909+ 1205882119889119909 = 2120582119906
2
1198672 120582 lt 0 (21)
We acquire
int
119878
1199062
119909(119905 119909) + 120588
2(119905 119909) 119889119909 le int
119878
1199062
0119909(119909) + 120588
2
0(119909) 119889119909 ≜ 119864
0
(22)
This completes the proof of Lemma 5
Lemma 6 Let 1198830= (1199060
1205880
) isin 119867119904times 119867119904minus1 119904 ge 2 and let 119879 be
the maximal existence time of the solution119883 = (119906
120588 ) to (3) withinitial data 119883
0 Then for all 119905 isin [0 119879) we have the following
results
int
119878
1199062(119905 119909) 119889119909 le 119890
1198622119905(int
119878
1199062
0(119909) 119889119909 + 1) (23)
where 1198621= max(|120590| minus 120582 1)119864
0+ sup
119905isin[0infin)|ℎ(119905)| gt 0 119862
2=
1198621+ 4119860 minus 2120582
Proof By computing directly we have
1003816100381610038161003816100381610038161003816
120597minus1
119909(
120590
2
1199062
119909+
1
2
1205882minus 119860119906 + 119892 + 120582119906
119909minus 120582120597minus1
119909119906) + ℎ (119905)
1003816100381610038161003816100381610038161003816
le int
119909
0
1003816100381610038161003816100381610038161003816
120590
2
1199062
119909+
1
2
1205882minus 119860119906 + 119892 + 120582119906
119909minus 120582120597minus1
119909119906
1003816100381610038161003816100381610038161003816
119889119909 + |ℎ (119905)|
le
1
2
max (|120590| minus 120582 1) 1198640+1003816100381610038161003816119892 (119905)
1003816100381610038161003816+ |ℎ (119905)|
+ (|119860| minus 120582) int
1
0
|119906| 119889119909 minus 120582
le max (|120590| minus 120582 1) 1198640+ |ℎ (119905)| + (2 |119860| minus 120582) int
119878
|119906| 119889119909 minus 120582
= 1198621+ (2 |119860| minus 120582) int
119878
|119906| 119889119909 minus 120582
(24)
where 1198621= max(|120590| minus 120582 1)119864
0+ sup119905isin[0infin)
|ℎ(119905)| gt 0 and
1003816100381610038161003816119892 (119905)
1003816100381610038161003816=
10038161003816100381610038161003816100381610038161003816
minus int
119878
(
120590
2
1199062
119909+
1
2
1205882minus 119860119906)119889119909
10038161003816100381610038161003816100381610038161003816
le
1
2
max (|120590| 1) 1198640+ 119860int
119878
|119906| 119889119909
(25)
4 Journal of Applied Mathematics
Multiplying (6) by 119906 and integrating with respect to 119909using the periodicity of 119906 and (24) we obtain
1
2
119889
119889119905
int
119878
1199062(119905 119909) 119889119909
= int
119878
119906119906119905119889119909
= minus120590int
119878
1199061199091199062119889119909
+ int
119878
119906 [120597minus1
119909(
120590
2
1199062
119909+
1
2
1205882minus 119860119906 + 120582119906
119909
minus 120582120597minus1
119909119906 + 119892) +ℎ (119905) ] 119889119909
= int
119878
119906 [120597minus1
119909(
120590
2
1199062
119909+
1
2
1205882minus 119860119906 + 120582119906
119909
minus120582120597minus1
119909119906 + 119892) + ℎ (119905) ] 119889119909
le (1198621+ (2 |119860| minus 120582) int
119878
|119906| 119889119909 minus 120582)int
119878
|119906| 119889119909
le 1198621int
119878
|119906| 119889119909 + (2 |119860| minus 120582) (int
119878
|119906| 119889119909)
2
minus 120582int
119878
1199062119889119909
le (
1198621
2
+ (2 |119860| minus 2120582))int
119878
1199062119889119909 +
1198621minus 120582
2
=
1198622
2
int
119878
1199062119889119909 +
1198621minus 120582
2
(26)
where 1198622= 1198621+ 4119860 minus 2120582 note that 119862
2gt 1198621
By Gronwallrsquos inequality we get
int
119878
1199062(119905 119909) 119889119909 le 119890
1198622119905(int
119878
1199062
0(119909) 119889119909 +
1198621minus 120582
1198622
) minus
1198621minus 120582
1198622
le 1198901198622119905(int
119878
1199062
0(119909) 119889119909 + 1)
(27)
This completes the proof of Lemma 6
Lemma 7 Assume that 1199060isin 119867119904(119878) 119904 ge 2 119906
0= 0 and that
the corresponding solution 119906(119905 119909) of (3) has a zero point forany time 119905 ge 0 Then for all 119905 isin [0 119879) we have
int
119878
1199062(119905 119909) 119889119909 le int
119878
1199062
119909(119905 119909) 119889119909 le 119864
0 (28)
Proof By assumption there is 1199090isin [0 1] such that 119906(119905 119909
0) =
0 for each 119905 isin [0 119879)Then for 119909 isin 119878 by holder equality we have
1199062(119905 119909) = (int
119909
1199090
119906119909119889119909)
2
le (119909 minus 1199090) int
119909
1199090
1199062
119909119889119909
119909 isin [1199090 1199090+
1
2
]
(29)
This implies sup119909isin119878
1199062(119905 119909) le (12) int
1198781199062
119909119889119909
int
119878
1199062(119905 119909) 119889119909 le sup
119909isin119878
1199062(119905 119909) le
1
2
int
119878
1199062
119909119889119909
le int
119878
1199062
119909+ 1205882119889119909 le 119864
0
(30)
3 Wave-Breaking Criteria
In this section by using transport equation theory we obtainthe wave-breaking criteria for solutions to (3) We first recallthe following propositions
Proposition 8 (1D Moser-type estimates) The following esti-mates hold
(a) For 119904 ge 010038171003817100381710038171198911198921003817100381710038171003817119867119904(R) le 119862 (
10038171003817100381710038171198911003817100381710038171003817119871infin(R)10038171003817100381710038171198921003817100381710038171003817119867119904(R) +
10038171003817100381710038171198911003817100381710038171003817119867119904(R)10038171003817100381710038171198921003817100381710038171003817119871infin(R)) (31)
(b) For 119904 gt 010038171003817100381710038171198911205971199091198921003817100381710038171003817119867119904(R) le 119862 (
10038171003817100381710038171198911003817100381710038171003817119871infin(R)10038171003817100381710038171205971199091198921003817100381710038171003817119867119904(R) +
10038171003817100381710038171198911003817100381710038171003817119867119904+1(R)10038171003817100381710038171198921003817100381710038171003817119871infin(R))
(32)
(c) For 1199041le 12 119904
2gt 12 119904
1+ 1199042gt 0
1003817100381710038171003817f11989210038171003817100381710038171198671199041 (R) le 119862
100381710038171003817100381711989110038171003817100381710038171198671199041 (R)
100381710038171003817100381711989210038171003817100381710038171198671199042 (R) (33)
where 1198621015840119904 are constants that are independent of 119891 and 119892
Proposition 9 (see [21]) Suppose that 119904 gt minus1198892 Let Vbe a vector field such that nablaV belongs to 119871
1([0 119879]119867
119904minus1) if
119904 gt 1 + 1198892 or to 1198711([0 119879]1198671198892 cap 119871infin) otherwise Suppose
also that 1198910
isin 119867119904 119865 isin 119871
1([0 119879]119867
119904) and that 119891 isin
119871infin([0 119879]119867
119904) cap 119862([0 119879] 119878
1015840) solves the 119889-dimensional linear
transport equations
(119879)
120597119905119891 + V sdot nabla119891 = 119865
1198911003816100381610038161003816119905=0
= 1198910
(34)
Then 119891 isin 119862([0 119879]119867119904) More precisely there exists a
constant119862 depending only on 119904119901 and119889 such that the followingstatements hold
(1) If 119904 = 1 + 1198892
10038171003817100381710038171198911003817100381710038171003817119867119904 le
10038171003817100381710038171198910
1003817100381710038171003817119867119904 + int
119905
0
119865 (120591)119867119904119889120591 + 119862int
119905
0
1198811015840(120591)
1003817100381710038171003817119891 (120591)
1003817100381710038171003817119867119904119889120591
(35)
or
10038171003817100381710038171198911003817100381710038171003817119867119904 le 119890119862119881(119905)
(10038171003817100381710038171198910
1003817100381710038171003817119867119904 + int
119905
0
119890minus119862119881(120591)
119865 (120591)119867119904119889120591) (36)
with 119881(119905) = int
119905
0nablaV(120591)
1198671198892cap119871infin119889120591 if 119904 lt 1 + 1198892 and 119881(119905) =
int
119905
0nablaV(120591)
119867119904minus1119889120591 else
Journal of Applied Mathematics 5
(2) If 119891 = V then for all 119904 gt 0 estimates (35) and (36) holdwith
119881 (119905) = int
119905
0
1003817100381710038171003817120597119909119906 (120591)
1003817100381710038171003817119871infin119889120591 (37)
Proposition 10 (see [21]) Let 0 lt 119904 lt 1 Suppose that 1198910isin
119867119904 119892 isin 119871
1([0 119879]119867
119904)V 120597119909V isin 119871
1([0 119879] 119871
infin) and that 119891 isin
119871infin([0 119879]119867
119904) cap 119862([0 119879] 119878
1015840) solves the 1-dimensional linear
transport equation
(119879)
120597119905119891 + V sdot nabla119891 = 119892
1198911003816100381610038161003816119905=0
= 1198910
(38)
Then 119891 isin 119862([0 119879]H119904) More precisely there exists a con-stant 119862 depending only on 119904 such that the following statementshold
10038171003817100381710038171198911003817100381710038171003817119867119904 le
10038171003817100381710038171198910
1003817100381710038171003817119867119904 + 119862int
119905
0
1003817100381710038171003817119892 (120591)
1003817100381710038171003817119867119904119889120591
+ 119862int
119905
0
1198811015840(120591)
1003817100381710038171003817119891 (120591)
1003817100381710038171003817119867119904119889120591
(39)
or
10038171003817100381710038171198911003817100381710038171003817119867119904 le 119890119862119881(119905)
(10038171003817100381710038171198910
1003817100381710038171003817119867119904 + 119862int
119905
0
1003817100381710038171003817119892 (120591)
1003817100381710038171003817119867119904119889120591) (40)
with 119881(119905) = int
119905
0(V(120591)
119871infin + 120597
119909V(120591)119871infin)119889120591
The above proposition was proved in [8] using Little-wood-Paley analysis for the transport equation and Moser-type estimates Using this result and performing the sameargument as in [17] we can obtain the following blow-upcriterion
Theorem 11 Let1198830= (1199060
1205880
) isin 119867119904times119867119904minus1 with 119904 ge 2 and119883 =
(119906
120588 ) be the corresponding solution to (3) Assume that 119879 gt 0 isthe maximal time of existence Then
119879 lt infin 997904rArr int
119879
0
1003817100381710038171003817120597119909119906 (120591)
1003817100381710038171003817119871infin119889120591 = infin (41)
Our next result describes the necessary and sufficientcondition for the blow-up of solutions to (3)
Theorem 12 Suppose that 120590 isin 119877 0 Let 1198830= (1199060
1205880
) isin 119867119904times
119867119904minus1 with 119904 ge 2 and let119879 be themaximal existence time of the
solution119883 = (119906
120588 ) to (3)with initial data1198830 Then the solutionblows up in finite time if and only if
lim inf119905rarr119879
minus
inf119909isin119878
120590119906119909(119905 119909) = minusinfin (42)
The approach one takes here is the method of charac-teristics Applying the following lemma we may carry outthe estimates along the characteristics 120593(119905 119909) which capturessup119909isin119878
119906119909(119905 119909) and inf
119909isin119878119906119909(119905 119909)
Lemma 13 (see [22]) Let119879 gt 0 and let V isin 1198621([0 119879]1198672(119877))Then for every 119905 isin [0 119879) there exists at least one point 120585(119905) isin 119877with
119898(119905) = inf119909isin119878
V119909(119905 119909) = V
119909(119905 120585 (119905)) (43)
and the function 119898(119905) is almost everywhere differentiable on(0 119879) with
119889119898 (119905)
119889119905
= V119905119909(119905 120585 (119905)) 119886119890 119900119899 (0 119879) (44)
Lemma 14 Let 1198830= (1199060
1205880
) isin 119867119904times 119867119904minus1 with 119904 ge 2 and let 119879
be the maximal existence time of the solution 119883 = (119906
120588 ) to (3)with initial data119883
0 Then one has the following
(1) 120590 = 0
sup119909isin119878
119906119909(119905 119909) le
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
+radic
10038171003817100381710038171205880
1003817100381710038171003817
2
119871infin(119878)+ 1198702
1(119879)
120590
+
1205822
1205902+
120582
120590
(120590 gt 0)
(45)
inf119909isin119878
119906119909(119905 119909) ge minus
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
minusradic1205822
1205902minus
1198702
2(119879)
120590
+
120582
120590
(120590 lt 0)
(46)
(2) 120590 = 0
sup119909isin119878
119906119909(119905 119909) le sup
119909isin119878
1199060119909
(119909)
+
1
2
(sup119909isin119878
1205882
0(119909) + 119870
2
1(119879))
119890120582119905minus 1
120582
(47)
inf119909isin119878
119906119909(119905 119909) ge inf
119909isin119878
1199060119909
(119909)
+
1
2
(inf119909isin119878
1205882
0(119909) minus 119870
2
2(119879))
119890120582119905minus 1
120582
(48)
The constants above are defined as follows
1198701(119879) = radic2119860 minus 120582 +
119860
2
1198640+
3119860 minus 2120582
2
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(49)
1198702(119879)
= radic2119860 minus 120582 +
119860 + 2
2
1198640+
3119860 minus 2120582
2
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(50)
Proof of Lemma 14 ByTheorem 2 and a simple density argu-ment we show that the desired results are valid when 119904 ge 3so we take 119904 = 3 in the proof
6 Journal of Applied Mathematics
Let 120590 gt 0 Using Lemma 13 and the fact that
sup119909isin119878
[V119909(119905 119909)] = minusinf
119909isin119878
[minusV119909(119905 119909)] (51)
We can consider119872(119905) and 120574(119905) as follows
119872(119905) = 119906119909(119905 120585 (119905)) = sup
119909isin119878
[119906119909(119905 119909)] 119905 isin [0 119879) (52)
Hence
119906119909119909(119905 120585 (119905)) = 0 ae on 119905 isin [0 119879) (53)
Take the trajectory 120593(119905 119909) defined in (12) Then we knowthat 120593(119905 sdot) 119877 rarr 119877 is a diffeomorphism for every 119905 isin [0 119879)Therefore there exists 119909
0(119905) isin 119877 such that
120593 (119905 1199090(119905)) = 120585 (119905) 119905 isin [0 119879) (54)
Now let
120574 (119905) = 120588 (119905 120593 (119905 1199090)) 119905 isin [0 119879) (55)
Therefore along the trajectory120593(119905 1199090) (4) and the second
equation of (3) become
1198721015840(119905) = minus
120590
2
1198722(119905) + 120582119872 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
1205741015840(119905) = minus120574119872 ae 119905 isin [0 119879)
(56)
where the notation denotes the derivative with respect to 119905
and 119891 represents the function
119891 = minus119860119906 minus 120582120597minus1
119909119906 + 119892 (119905)
= minus119860119906 minus 120582120597minus1
119909119906 minus int
119878
(
120590
2
1199062
119909+
1
2
1205882minus 119860119906)119889119909
(57)
We first compute the upper and lower bounds for 119891 forlater use in getting the blow-up result as follows
119891 = minus119860119906 minus 120582int
119909
0
119906 119889119909 minus
120590
2
int
119878
1199062
119909119889119909 minus
1
2
int
119878
1205882119889119909 + 119860int
119878
119906 119889119909
le minus119860119906 minus 120582int
119909
0
119906 119889119909 + 119860int
119878
119906 119889119909 le
119860
2
(1 + 1199062)
+
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
(58)
Since 1199062 le (12) int119878(1199062+ 1199062
119909)119889119909 (17) we obtain the upper
bound for 119891
119891 le
119860
2
(1 +
1
2
int
119878
(1199062+ 1199062
119909) 119889119909) +
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le 119860 minus
120582
2
+
119860
4
int
119878
(1205882+ 1199062
119909) 119889119909 +
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860
4
1198640+
3119860 minus 2120582
4
[1198901198622119905(int
119878
1199062
0(119909) 119889119909 + 1)]
le 119860 minus
120582
2
+
119860
4
1198640+
3119860 minus 2120582
4
times [1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)] =
1
2
1198702
1(119879)
(59)
Now we turn to the lower bound of 119891 Using previous argu-ments we get
minus119891 = 119860119906 + 120582int
119909
0
119906 119889119909 +
120590
2
int
119878
1199062
119909119889119909 +
1
2
int
119878
1205882119889119909 minus 119860int
119878
119906 119889119909
le 119860
1 + 1199062
2
+
max (|120590| 1)2
int
119878
(1205882+ 1199062
119909) 119889119909
+
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le 119860 minus
120582
2
+
119860 + 2max (|120590| 1)4
int
119878
(1205882+ 1199062
119909) 119889119909
+
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860 + 2max (|120590| 1)4
1198640
+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(60)
When 120590 lt 0 we have a finer estimate
minus119891 le 119860 minus
120582
2
+
119860 + 2
4
int
119878
(1205882+ 1199062
119909) 119889119909 +
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860 + 2
4
1198640+
3119860 minus 2120582
4
times [1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)] =
1
2
1198702
2(119879)
(61)
Combining (59) and (60) we obtain
10038161003816100381610038161198911003816100381610038161003816le 119860 minus
120582
2
+
119860 + 2max (|120590| 1)4
1198640
+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)] =
1
2
1198702
3(119879)
(62)
Since 119904 ge 3 we have 119906 isin 11986210(119878) Therefore
sup119909isin119878
119906119909(119905 119909) ge 0 inf
119909isin119878
119906119909(119905 119909) le 0 119905 isin [0 119879) (63)
Journal of Applied Mathematics 7
Hence119872(119905) gt 0 for 119905 isin [0 119879) From the second equationof (55) we obtain
120574 (119905) = 120574 (0) 119890minusint119905
0
119872(120591)119889120591 (64)
Hence1003816100381610038161003816120588 (119905 120593 (119905 119909
0))1003816100381610038161003816=1003816100381610038161003816120574 (119905)
1003816100381610038161003816le1003816100381610038161003816120574 (0)
1003816100381610038161003816le10038171003817100381710038171205880
1003817100381710038171003817119871infin(119878) (65)
For any given 119909 isin 119878 define
1199011(119905) = 119872 (119905) minus
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
minusradic
10038171003817100381710038171205880
1003817100381710038171003817
2
119871infin(119878)+ 1198702
1(119879)
120590
+
1205822
1205902minus
120582
120590
(120590 gt 0)
(66)
Observing that 1199011(119905) is a 1198621-differentiable function on
[0 119879) and satisfies
1199011(0) = 119872 (0) minus
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
minusradic
10038171003817100381710038171205880
1003817100381710038171003817
2
119871infin(119878)+ 1198702
1(119879)
120590
+
1205822
1205902minus
120582
120590
le 119872(0) minus10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
(67)
We now claim that 1199011(119905) le 0 119905 isin [0 119879)
Assume the contrary that there is 1199050isin [0 119879) such that
1199011(1199050) gt 0
Let 1199051= max119905 lt 119905
0 1199011(119905) = 0 Then 119901
1(1199051) = 0 and
1199011015840
1(1199051) ge 0 or equivalently
119872(1199051) =
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)+radic
10038171003817100381710038171205880
1003817100381710038171003817
2
119871infin(119878)+ 1198702
1(119879)
120590
+
1205822
1205902+
120582
120590
(68)
and1198721015840(1199051) ge 0 ae 119905 isin [0 119879) On the other hand we have
1198721015840(1199051) = minus
120590
2
1198722(1199051) + 120582119872(119905
1) +
1
2
1205742(1199051)
+ 119891 (1199051 120593 (1199051 1199090)) ae 119905 isin [0 119879)
le minus
120590
2
(119872(1199051) minus
120582
120590
)
2
+
1
2
10038171003817100381710038171205880
1003817100381710038171003817
2
119871infin(119878)+
1205822
2120590
+
1198702
1(119879)
2
lt 0
(69)
which is a contradictionTherefore1199011(119905) le 0 for all 119905 isin [0 119879)
Since 119909 is arbitrarily chosen we obtain (45)To derive (46) in the case of 120590 lt 0 we consider (119905) and
120574(119905) as in Lemma 13
(119905) = 119906119909(119905 120577 (119905)) = inf
119909isin119878
[119906119909(119905 119909)] 119905 isin [0 119879) (70)
Hence
119906119909119909(119905 120577 (119905)) = 0 ae 119905 isin [0 119879) (71)
Using previous arguments we take the characteristic 120593(119905119909) defined in (13) and choose 119909
1(119905) isin 119877 such that
120593 (119905 1199091(119905)) = 120577 (119905) (72)
Let
120574 (119905) = 120588 (119905 120593 (119905 119909)) 119905 isin [0 119879) (73)
Hence along the trajectory 1015840(119905) = 120582(119905) +(12)1205742(119905) +
119891(119905 120593(119905 1199090)) ge 120582(0)+(12)120574
2(0) +(12)119870
2
2(119879) (4) and the
second equation of (3) become
1015840(119905) = minus
120590
2
2(119905) + 120582 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
1205741015840(119905) = minus120574 ae 119905 isin [0 119879)
(74)
Define
1199012(119905) = (119905) +
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
+radic1205822
1205902minus
1198702
2(119879)
120590
minus
120582
120590
(120590 lt 0)
(75)
For any given 119909 isin 119878 Note that 1199012(119905) is also 119862
1-dif-ferentiable function on [0 119879) and satisfies
1199012(0) = (0) +
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
+radic1205822
1205902minus
1198702
2(119879)
120590
minus
120582
120590
ge (0) +10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
ge 0
(76)
We now claim that 1199012(119905) ge 0 for any 119905 isin [0 119879)
Suppose not then there is isin [0 119879) such that 1199012(1199050) lt 0
Define
1199052= max 119905 lt 119905
0 1199012(119905) = 0 (77)
Then 1199012(1199052) = 0 and 1199011015840
2(1199052) lt 0 or equivalently
(1199052) = minus
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)minusradic1205822
1205902minus
1198702
2(119879)
120590
+
120582
120590
(78)
and 1015840(1199052) le 0 ae 119905 isin [0 119879) However we have
1015840(1199052) = minus
120590
2
2(1199052) + 120582 (119905
2) +
1
2
1205742(1199052)
+ 119891 (1199052 120593 (1199052 1199090)) ae 119905 isin [0 119879)
ge minus
120590
2
(minus10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)minusradic1205822
1205902minus
1198702
2(119879)
120590
)
2
+
1205822
2120590
minus
1
2
1198702
2(119879) gt 0
(79)
8 Journal of Applied Mathematics
Therefore 1199012(119905) ge 0 for any 119905 isin [0 119879) Since 119909 is chosen
arbitrarily we obtain (46)Let 120590 = 0 Using previous arguments (56) becomes
1198721015840(119905) = minus
120590
2
1198722(119905) + 120582119872 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
1205741015840(119905)
= minus120574119872 ae 119905 isin [0T) (80)
where the notation denotes the derivative with respect to 119905
and 119891 represents the function
119891 = minus119860119906 minus 120582120597minus1
119909119906 + 119892 (119905)
= minus119860119906 minus 120582120597minus1
119909119906 minus int
119878
(
1
2
1205882minus 119860119906)119889119909
(81)
We first compute the upper and lower bounds for 119891 forlater use in getting the blow-up result
119891 = minus119860119906 minus 120582int
119909
0
119906 119889119909 minus
1
2
int
119878
1205882119889119909 + 119860int
119878
119906 119889119909
le minus119860119906 minus 120582int
119909
0
119906 119889119909 + 119860int
119878
119906 119889119909 le
119860
2
(1 + 1199062)
+
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le
119860
2
(1 +
1
2
int
119878
(1199062+ 1199062
119909) 119889119909) +
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le 119860 minus
120582
2
+
119860
4
int
119878
(1205882+ 1199062
119909) 119889119909 +
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860
4
1198640+
3119860 minus 2120582
4
[1198901198622119905(int
119878
1199062
0(119909) 119889119909 + 1)]
le 119860 minus
120582
2
+
119860
4
1198640+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(82)
Now we turn to the lower bound of 119891
minus119891 = 119860119906 + 120582int
119909
0
119906 119889119909 +
1
2
int
119878
1205882119889119909 minus 119860int
119878
119906 119889119909
le 119860
1 + 1199062
2
+
1
2
int
119878
1205882119889119909 +
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le 119860 minus
120582
2
+
119860 + 2
4
int
119878
(1205882+ 1199062
119909) 119889119909 +
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860 + 2
4
1198640+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(83)
Combining (82) and (83) we obtain
10038161003816100381610038161198911003816100381610038161003816le 119860 minus
120582
2
+
119860 + 2
4
1198640+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(84)
We know 119872(119905) gt 0 for 119905 isin [0 119879) From the secondequation of (81) we obtain that
120574 (119905) = 120574 (0) 119890minusint119905
0
119872(120591)119889120591 (85)
Hence1003816100381610038161003816120588 (119905 120593 (119905 119909
0))1003816100381610038161003816=1003816100381610038161003816120574 (119905)
1003816100381610038161003816le1003816100381610038161003816120574 (0)
1003816100381610038161003816 (86)
Therefore we have
1198721015840(119905) = 120582119872 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
le 120582119872 (0) +
1
2
1205742(0) +
1
2
1198702
1(119879)
(87)
1198721015840(119905) minus 120582119872 (119905) le
1
2
(sup119909isin119878
1205882
0(119909) + 119870
2
1(119879)) ae 119905 isin [0 119879)
(88)
Integrating (88) on [0 119905] we prove (47) as follows
119872(119905) le sup119909isin119878
1199060119909
(119909) +
1
2
(sup119909isin119878
1205882
0(119909) + 119870
2
1(119879))
119890120582119905minus 1
120582
(89)
To obtain a lower bound for inf119909isin119878
119906119909(119905 119909) we use the
same argumentSince 120590 = 0 (80) becomes
1015840(119905) = 120582 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
1))
1205741015840(119905) = minus120574 ae 119905 isin [0 119879)
(90)
Because of (119905) lt 0 we get from the second equation of(90) that
120574 (119905) = 120574 (0) 119890minusint119905
0
119872(120591)119889120591 (91)
This means that1003816100381610038161003816120588 (119905 120593 (119905 119909
1))1003816100381610038161003816=1003816100381610038161003816120574 (119905)
1003816100381610038161003816ge1003816100381610038161003816120574 (0)
1003816100381610038161003816 (92)
Then
1015840(119905) = 120582 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
ge 120582 (0) +
1
2
1205742(0) +
1
2
1198702
2(119879)
(93)
1015840(119905) minus 120582 (119905) ge
1
2
(inf119909isin119878
1205882
0(119909) minus 119870
2
2(119879)) ae 119905 isin [0 119879)
(94)
Integrating (94) on [0 119905] we prove (48) This completesthe proof of Lemma 14
Lemma 15 Suppose that 120590 isin 119877 0 Suppose 1198830= (1199060
1205880
) isin
119867119904times119867119904minus1 with 119904 ge 2 and let 119879 be the maximal existence time
Journal of Applied Mathematics 9
of the solution 119883 = (119906
120588 ) to (3) with initial data 1198830 Then we
have
120588 (119905 120593 (119905 119909)) 120593119909(119905 119909) = 120588
0(119909) (119905 119909) isin [0 119879] times 119878 (95)
Moreover if there exists119872 gt 0 such that
inf(119905119909)isin[0119879]times119878
120590119906119909(119905 119909) ge minus119872 (119905 119909) isin [0 119879] times 119878 (96)
Then1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin(119878)
=1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin(119878)
le 119890119872119879120590 1003817
1003817100381710038171205880
1003817100381710038171003817119871infin(119878)
(120590 gt 0)
(97)
1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin(119878)
=1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin(119878)
le 119890119873119879120590 1003817
1003817100381710038171205880
1003817100381710038171003817119871infin(119878)
(120590 lt 0)
(98)
where119873 = 1199060119909119871infin(119878)+radic12058221205902minus 1198702
2(119879)120590 minus 120582120590 and119870
2(119879)
is given in (50)
Proof Differentiating the left hand side of (95) with respectto 119905 in view of the relations (12) and (3) we obtain
119889
119889119905
120588 (119905 120593 (119905 119909)) 120593119909(119905 119909)
= [120588119905(119905 120593) + 120588
119909(119905 120593) 120593
119905(119905 119909)]
times 120593119909(119905 119909) + 120588 (119905 120593) 120593
119909119905(119905 119909)
= [120588119905(119905 120593) + 120588
119909(119905 120593) 120593
119905(119905 119909)] 120593
119909(119905 119909)
+ 120588 (119905 120593) 119906119909(119905 120593) 120593
119909(119905 119909)
= [120588119905(119905 120593) + 120588
119909(119905 120593) 120593
119905(119905 119909) + 120588 (119905 120593) 119906
119909(119905 120593)]
times 120593119909(119905 119909) = 0
(99)
This completes the proof of (95) In view of the assump-tion (96) and 120590 gt 0 we obtain 119906(119905 119909) ge minus(119872120590) (119905 119909) isin
[0 119879] times 119878By Lemma 3 and (95) we have
1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin =
1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin =
1003817100381710038171003817100381710038171003817
119890minusint119905
0
119906119909(120591sdot)119889120591
1205880(sdot)
1003817100381710038171003817100381710038171003817119871infin
le 1198901198721198791205901003817
1003817100381710038171205880(sdot)1003817100381710038171003817119871infin
(100)
To obtain (98) we use a similar argument as before Using(13) and the lower bound for 119906
119909(119905 119909) in (46) it follows that
1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin =
1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin =
1003817100381710038171003817100381710038171003817
119890minusint119905
0
119906119909(120591sdot)119889120591
1205880(sdot)
1003817100381710038171003817100381710038171003817119871infin
le 1198901198731198791205901003817
1003817100381710038171205880(sdot)1003817100381710038171003817119871infin
(101)
which proves (98) This completes the proof of Lemma 15
Proof of Theorem 12 Suppose that119879 lt infin and that (42) is notvalid Then there is some positive number119872 gt 0 such that
120590119906119909(119905 119909) ge minus119872 (119905 119909) isin [0 119879] times 119878 (102)
It now follows from Lemma 14 that |119906119909(119905 119909)| le 119862 where
119862 = 119862(119860119872 120590 1198640 120582 119906
0 119879) Therefore Theorem 11 implies
that the maximal existence time 119879 = infin which contradictswith the assumption that 119879 lt infin
Conversely the Sobolev embedding theorem 119867119904(119878) rarr
119871infin(119878) with 119904 gt 12 implies that if (70) holds the corre-
sponding solution blows up in finite time which completesthe proof of Theorem 12
Acknowledgments
The authors would like to thank the referee for commentsand suggestions This work is supported by the NationalNature Science Foundation of China (no 11171135) theNature Science Foundation of the Jiangsu Higher EducationInstitutions of China (no 09KJB110003) and the high-leveltalented person special subsidizes of Jiangsu University (no05JDG047)
References
[1] J K Hunter and R Saxton ldquoDynamics of director fieldsrdquo SIAMJournal on Applied Mathematics vol 51 no 6 pp 1498ndash15211991
[2] Z Yin ldquoOn the structure of solutions to the periodic Hunter-Saxton equationrdquo SIAM Journal on Mathematical Analysis vol36 no 1 pp 272ndash283 2004
[3] H-H Dai and M Pavlov ldquoTransformations for the Camassa-Holm equation its high-frequency limit and the Sinh-Gordonequationrdquo Journal of the Physical Society of Japan vol 67 no 11pp 3655ndash3657 1998
[4] J K Hunter and Y X Zheng ldquoOn a completely integrablenonlinear hyperbolic variational equationrdquo Physica D vol 79no 2ndash4 pp 361ndash386 1994
[5] R Camassa and D D Holm ldquoAn integrable shallow waterequation with peaked solitonsrdquo Physical Review Letters vol 71no 11 pp 1661ndash1664 1993
[6] R S Johnson ldquoCamassa-Holm Korteweg-de Vries and relatedmodels for water wavesrdquo Journal of Fluid Mechanics vol 455pp 63ndash82 2002
[7] A Constantin and B Kolev ldquoOn the geometric approach to themotion of inertialmechanical systemsrdquo Journal of Physics A vol35 no 32 pp R51ndashR79 2002
[8] P J Olver and P Rosenau ldquoTri-Hamiltonian duality betweensolitons and solitary-wave solutions having compact supportrdquoPhysical Review E vol 53 no 2 pp 1900ndash1906 1996
[9] R Beals D H Sattinger and J Szmigielski ldquoInverse scatteringsolutions of the Hunter-Saxton equationrdquo Applicable Analysisvol 78 no 3-4 pp 255ndash269 2001
[10] A Bressan and A Constantin ldquoGlobal solutions of the Hunter-Saxton equationrdquo SIAM Journal on Mathematical Analysis vol94 pp 68ndash92 2010
[11] A Bressan H Holden and X Raynaud ldquoLipschitz metric forthe Hunter-Saxton equationrdquo Journal de Mathematiques Pureset Appliquees vol 94 no 1 pp 68ndash92 2010
10 Journal of Applied Mathematics
[12] J K Hunter and Y X Zheng ldquoOn a nonlinear hyperbolicvariational equation I Global existence of weak solutionsrdquoArchive for Rational Mechanics and Analysis vol 129 no 4 pp305ndash353 1995
[13] J Lenells ldquoThe Hunter-Saxton equation describes the geodesicflow on a sphererdquo Journal of Geometry and Physics vol 57 no10 pp 2049ndash2064 2007
[14] J B Li and Y S Li ldquoBifurcations of travelling wave solutions fora two-component Camassa-Holm equationrdquoActaMathematicaSinica (English Series) vol 24 no 8 pp 1319ndash1330 2008
[15] W Rui and Y Long ldquoIntegral bifurcation method together witha translation-dilation transformation for solving an integrable2-component Camassa-Holm shallow water systemrdquo Journal ofApplied Mathematics vol 2012 Article ID 736765 21 pages2012
[16] H Wu and M Wunsch ldquoGlobal existence for the generalizedtwo-component Hunter-Saxton systemrdquo Journal of Mathemati-cal Fluid Mechanics vol 14 no 3 pp 455ndash469 2012
[17] BMoon andY Liu ldquoWave breaking and global existence for thegeneralized periodic two-component Hunter-Saxton systemrdquoJournal of Differential Equations vol 253 no 1 pp 319ndash3552012
[18] S Wu and Z Yin ldquoBlow-up blow-up rate and decay of thesolution of the weakly dissipative Camassa-Holm equationrdquoJournal ofMathematical Physics vol 47 no 1 Article ID 0135042006
[19] S Wu J Escher and Z Yin ldquoGlobal existence and blow-up phenomena for a weakly dissipative Degasperis-Procesiequationrdquo Discrete and Continuous Dynamical Systems B vol12 no 3 pp 633ndash645 2009
[20] T Kato ldquoQuasi-linear equations of evolution with applicationsto partial differential equationsrdquo in Spectral Theory and Differ-ential Equations vol 448 of Lecture Notes in Mathematics pp25ndash70 Springer Berlin Germany 1975
[21] G Gui and Y Liu ldquoOn the global existence and wave-breakingcriteria for the two-component Camassa-Holm systemrdquo Journalof Functional Analysis vol 258 no 12 pp 4251ndash4278 2010
[22] A Constantin and J Escher ldquoWave breaking for nonlinearnonlocal shallow water equationsrdquo Acta Mathematica vol 181no 2 pp 229ndash243 1998
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Applied Mathematics
solutions of the weakly dissipative CH equation and DPequation Inspired by the results mentioned above we aregoing to discuss the initial value problem associated withthe generalized weakly dissipative periodic two-componentHunter-Saxton system
119906119905119909119909
+ 2120590119906119909119906119909119909+ 120590119906119906
119909119909119909minus 120588120588119909+ 119860119906119909
+ 120582 (119906 minus 119906119909119909) = 0 119905 gt 0 119909 isin 119877
120588119905+ (120588119906)
119909= 0 119905 gt 0 119909 isin 119877
119906 (119905 119909 + 1) = 119906 (119905 119909) 120588 (119905 119909 + 1) = 120588 (119905 119909)
119905 ge 0 119909 isin 119877
119906 (0 119909) = 1199060(119909) 120588 (0 119909) = 120588
0(119909) 119909 isin 119877
(3)
where 120590 isin 119877 is the new free parameter and 119860 ge 0 120582 lt 0Our major results of this paper are Theorems 11 and
12 (wave-breaking criterion) The remainder of the paperis organized as follows Section 2 establishes the local well-posedness for (3) with the initial data in 119867
119904times 119867119904minus1 119904 ge 2
Section 3 deals with the wave breaking of this new systemTheorem 11 using transport equation theory states a wave-breaking criterion which says that the wave breaking onlydepends on the slope of 119906 not the slope of 120588 Theorem 12improves the blow-up criterion with a more precise condi-tion
Notation Throughout this paper 119878 = 119877119885 will denote theunit circle By119867119904 119904 ge 0 we will represent the Sobolev spacesof equivalence classes of functions defined on the unit circle119878 which have square-integrable distributional derivatives upto order 119904 The119867119904-norm will be designated by sdot
119867119904 and the
norm of a vector V isin 119867119904 times 119867119904minus1 will be written as V119867119904times119867119904minus1
Also the Lebesgue spaces of order 119901 isin [1infin]will be denotedby 119871119901(119878) and the norm of their elements will be denoted
by 119891119871119901(119878) Finally if 119901 = 2 we agree on the convention
sdot 1198712(119878)
= sdot
2 Preliminaries
In this part we will establish the local well-posedness forthe Cauchy problem of system (3) by using Katorsquos theory Topursue our goal we give the results we wanted in brief
We now provide the framework in which we will refor-mulate (3) To do this we observe that we can write the firstequation of (3) in the following integrated form
119906119905119909+
120590
2
1199062
119909+ 120590119906119906
119909119909minus
1
2
1205882+ 119860119906 + 120582120597
minus1
119909119906 minus 120582119906
119909= 119892 (119905)
(4)
where 120597minus1119909119891(119909) = int
119909
0119891(119910)119889119910 and 119892(119905) is determined by the
periodicity of 119906 to be
119892 (119905) = minusint
119878
(
120590
2
1199062
119909+
1
2
1205882minus 119860119906)119889119909 (5)
Integrating both sides of (4) with respect to variable 119909we get
119906119905+ 120590119906119906
119909= 120597minus1
119909(
120590
2
1199062
119909+
1
2
1205882minus 119860119906 + 119892 + 120582119906
119909minus 120582120597minus1
119909119906)
+ ℎ (119905)
(6)
where ℎ(119905) [0infin) rarr 119877 is an arbitrary continuous functionTherefore (3) can be written in the ldquotransportrdquo form asfollows
119906119905+ 120590119906119906
119909= 120597minus1
119909(
120590
2
1199062
119909+
1
2
1205882minus 119860119906 minus 120582120597
minus1
119909119906 + 120582119906
119909+ 119892)
+ ℎ (119905) 119905 gt 0 119909 isin 119877
120588119905+ 119906120588119909= minus119906119909120588 119905 gt 0 119909 isin 119877
119906 (119905 119909 + 1) = 119906 (119905 119909) 120588 (119905 119909 + 1) = 120588 (119905 119909)
119905 ge 0 119909 isin 119877
119906 (0 119909) = 1199060(119909) 120588 (0 119909) = 120588
0(119909) 119909 isin 119877
(7)
where ℎ(119905) [0infin) rarr 119877 is an arbitrary continuous functionNext we apply Katorsquos theory to establish the local well-
posedness for the system (3) Consider the abstract quasi-linear evolution equation
119889V
119889119905
+ 119860 (V) V = 119891 (V) 119905 ge 0 V (0) = V0 (8)
Proposition 1 (see [20]) Given the evolution equation (8)assume that the Kato conditions hold For a fixed V
0isin 119884 there
is a maximal 119879 gt 0 depending only on V0119884and a unique
solution V to the abstract quasi-linear evolution equation (8)such that
V = V (sdot V0) isin 119862 ([0 119879) 119884) cap 119862
1([0 119879) 119883) (9)
Moreover the map V0rarr V(sdot V
0) is continuous from 119884 to
119862 ([0 119879) 119884) cap 1198621([0 119879) 119883) (10)
One may follow the similar argument as in [17] to obtain thefollowing local well-posedness for (3)
Theorem 2 Given any 1198830= (1199060
1205880
) isin 119867119904times 119867119904minus1 119904 ge 2 there
exist a maximal 119879 = 119879(120590 119860 1198830119867119904times119867119904minus1) gt 0 and a unique
solution119883 = (119906
120588 ) to (3) such that
119883 = 119883 (sdot 1198830) isin 119862 ([0 119879) 119867
119904(119878) times 119867
sminus1(119878))
cap 1198621([0 119879) 119867
119904minus1(119878) times 119867
119904minus2(119878))
(11)
Moreover the solution depends continuously on the initialdata that is the mapping 119883
0rarr 119883(sdot 119883
0) 119867
119904times 119867119904minus1
rarr
119862([0 119879)119867119904(119878) times 119867
119904minus1(119878)) cap 119862
1([0 119879) and 119867
119904minus1(119878) times
119867119904minus2(119878)) is continuous and the maximal existence time 119879 can
be chosen independently of the Sobolev order 119904
Journal of Applied Mathematics 3
Now discuss the initial value problem for the Lagrangian flowmap as follows
120597120593
120597119905
= 119906 (119905 120593 (119905 119909)) 119905 isin [0 119879)
120593 (0 119909) = 119909 119909 isin 119877
(12)
where 119906 is the first component of the solution 119883 to (3) Usingclassical results from ordinary differential equations one canacquire the following result on 120593which is of vital importance inthe proof of the blow-up scenarios
Lemma 3 (see [17]) Let 119906 isin 119862([0 119879)119867119904) cap 1198621([0 119879)119867119904minus1)119904 ge 2 Then initial value problem (12) admits a unique solu-tion 120593 isin 119862
1([0 119879) times 119877 119877) Moreover 120593(119905 sdot)
119905isin[0119879)is increas-
ing diffeomorphism of 119877 with
120593119909(119905 119909) = 119890
int119879
0
119906119909(120591120593(120591119909))119889120591
gt 0 (119905 119909) isin [0 119879) times 119877 (13)
Remark 4 Since 120593(119905 sdot) 119877 rarr 119877 is a diffeomorphism ofthe linear for every 119905 isin [0 119879) the 119871infin-norm of any functionV(119905 sdot) isin 119871
infin 119905 isin [0 119879) is preserved under the family ofdiffeomorphisms 120593(119905 sdot) with 119905 isin [0 119879) that is
V (119905 sdot)119871infin(119878)
=1003817100381710038171003817V (119905 120593 (119905 sdot))100381710038171003817
1003817119871infin(119878) 119905 isin [0 119879) (14)
Similarly we have
inf119909isin119878
V (119905 119909) = inf119909isin119878
V (119905 120593 (119905 119909)) 119905 isin [0 119879)
sup119909isin119878
V (119905 119909) = sup119909isin119878
V (119905 120593 (119905 119909)) 119905 isin [0 119879)
(15)
Lemma 5 Let 1198830= (1199060
1205880
) isin 119867119904times 119867119904minus1 119904 ge 2 and let 119879 be
the maximal existence time of the solution119883 = (119906
120588 ) to (3) withinitial data 119883
0 Then for all 119905 isin [0 119879) we have the following
results
int
119878
120588 (119905 119909) 119889119909 = int
119878
1205880(119905) 119889119909 (16)
int
119878
1199062
119909(119905 119909) + 120588
2(119905 119909) 119889119909 le int
119878
1199062
0119909(119909) + 120588
2
0(119909) 119889119909 ≜ 119864
0
(17)
Proof On the one hand integrating the second equation in(3) by parts and using the periodicity of 119906 and 120588 we acquire
119889
119889119905
int
119878
120588 119889119909 = minusint
119878
(119906120588)119909119889119909 = 0 (18)
On the other hand multiplying (4) by 119906119909and integrating
by parts considering the periodicity of 119906 we obtain
119889
119889119905
int
119878
1199062
119909119889119909 = minus2int
119878
119906120588120588119909119889119909 + 2120582int
119878
1199062119889119909 + 2120582int
119878
1199062
119909119889119909
(19)
Multiplying the second equation in (3) by 120588 and integrat-ing by parts we have
119889
119889119905
int
119878
1205882119889119909 = 2int
119878
119906120588120588119909119889119909 (20)
Adding the above two equations we get
119889
119889119905
int
119878
1199062
119909+ 1205882119889119909 = 2120582119906
2
1198672 120582 lt 0 (21)
We acquire
int
119878
1199062
119909(119905 119909) + 120588
2(119905 119909) 119889119909 le int
119878
1199062
0119909(119909) + 120588
2
0(119909) 119889119909 ≜ 119864
0
(22)
This completes the proof of Lemma 5
Lemma 6 Let 1198830= (1199060
1205880
) isin 119867119904times 119867119904minus1 119904 ge 2 and let 119879 be
the maximal existence time of the solution119883 = (119906
120588 ) to (3) withinitial data 119883
0 Then for all 119905 isin [0 119879) we have the following
results
int
119878
1199062(119905 119909) 119889119909 le 119890
1198622119905(int
119878
1199062
0(119909) 119889119909 + 1) (23)
where 1198621= max(|120590| minus 120582 1)119864
0+ sup
119905isin[0infin)|ℎ(119905)| gt 0 119862
2=
1198621+ 4119860 minus 2120582
Proof By computing directly we have
1003816100381610038161003816100381610038161003816
120597minus1
119909(
120590
2
1199062
119909+
1
2
1205882minus 119860119906 + 119892 + 120582119906
119909minus 120582120597minus1
119909119906) + ℎ (119905)
1003816100381610038161003816100381610038161003816
le int
119909
0
1003816100381610038161003816100381610038161003816
120590
2
1199062
119909+
1
2
1205882minus 119860119906 + 119892 + 120582119906
119909minus 120582120597minus1
119909119906
1003816100381610038161003816100381610038161003816
119889119909 + |ℎ (119905)|
le
1
2
max (|120590| minus 120582 1) 1198640+1003816100381610038161003816119892 (119905)
1003816100381610038161003816+ |ℎ (119905)|
+ (|119860| minus 120582) int
1
0
|119906| 119889119909 minus 120582
le max (|120590| minus 120582 1) 1198640+ |ℎ (119905)| + (2 |119860| minus 120582) int
119878
|119906| 119889119909 minus 120582
= 1198621+ (2 |119860| minus 120582) int
119878
|119906| 119889119909 minus 120582
(24)
where 1198621= max(|120590| minus 120582 1)119864
0+ sup119905isin[0infin)
|ℎ(119905)| gt 0 and
1003816100381610038161003816119892 (119905)
1003816100381610038161003816=
10038161003816100381610038161003816100381610038161003816
minus int
119878
(
120590
2
1199062
119909+
1
2
1205882minus 119860119906)119889119909
10038161003816100381610038161003816100381610038161003816
le
1
2
max (|120590| 1) 1198640+ 119860int
119878
|119906| 119889119909
(25)
4 Journal of Applied Mathematics
Multiplying (6) by 119906 and integrating with respect to 119909using the periodicity of 119906 and (24) we obtain
1
2
119889
119889119905
int
119878
1199062(119905 119909) 119889119909
= int
119878
119906119906119905119889119909
= minus120590int
119878
1199061199091199062119889119909
+ int
119878
119906 [120597minus1
119909(
120590
2
1199062
119909+
1
2
1205882minus 119860119906 + 120582119906
119909
minus 120582120597minus1
119909119906 + 119892) +ℎ (119905) ] 119889119909
= int
119878
119906 [120597minus1
119909(
120590
2
1199062
119909+
1
2
1205882minus 119860119906 + 120582119906
119909
minus120582120597minus1
119909119906 + 119892) + ℎ (119905) ] 119889119909
le (1198621+ (2 |119860| minus 120582) int
119878
|119906| 119889119909 minus 120582)int
119878
|119906| 119889119909
le 1198621int
119878
|119906| 119889119909 + (2 |119860| minus 120582) (int
119878
|119906| 119889119909)
2
minus 120582int
119878
1199062119889119909
le (
1198621
2
+ (2 |119860| minus 2120582))int
119878
1199062119889119909 +
1198621minus 120582
2
=
1198622
2
int
119878
1199062119889119909 +
1198621minus 120582
2
(26)
where 1198622= 1198621+ 4119860 minus 2120582 note that 119862
2gt 1198621
By Gronwallrsquos inequality we get
int
119878
1199062(119905 119909) 119889119909 le 119890
1198622119905(int
119878
1199062
0(119909) 119889119909 +
1198621minus 120582
1198622
) minus
1198621minus 120582
1198622
le 1198901198622119905(int
119878
1199062
0(119909) 119889119909 + 1)
(27)
This completes the proof of Lemma 6
Lemma 7 Assume that 1199060isin 119867119904(119878) 119904 ge 2 119906
0= 0 and that
the corresponding solution 119906(119905 119909) of (3) has a zero point forany time 119905 ge 0 Then for all 119905 isin [0 119879) we have
int
119878
1199062(119905 119909) 119889119909 le int
119878
1199062
119909(119905 119909) 119889119909 le 119864
0 (28)
Proof By assumption there is 1199090isin [0 1] such that 119906(119905 119909
0) =
0 for each 119905 isin [0 119879)Then for 119909 isin 119878 by holder equality we have
1199062(119905 119909) = (int
119909
1199090
119906119909119889119909)
2
le (119909 minus 1199090) int
119909
1199090
1199062
119909119889119909
119909 isin [1199090 1199090+
1
2
]
(29)
This implies sup119909isin119878
1199062(119905 119909) le (12) int
1198781199062
119909119889119909
int
119878
1199062(119905 119909) 119889119909 le sup
119909isin119878
1199062(119905 119909) le
1
2
int
119878
1199062
119909119889119909
le int
119878
1199062
119909+ 1205882119889119909 le 119864
0
(30)
3 Wave-Breaking Criteria
In this section by using transport equation theory we obtainthe wave-breaking criteria for solutions to (3) We first recallthe following propositions
Proposition 8 (1D Moser-type estimates) The following esti-mates hold
(a) For 119904 ge 010038171003817100381710038171198911198921003817100381710038171003817119867119904(R) le 119862 (
10038171003817100381710038171198911003817100381710038171003817119871infin(R)10038171003817100381710038171198921003817100381710038171003817119867119904(R) +
10038171003817100381710038171198911003817100381710038171003817119867119904(R)10038171003817100381710038171198921003817100381710038171003817119871infin(R)) (31)
(b) For 119904 gt 010038171003817100381710038171198911205971199091198921003817100381710038171003817119867119904(R) le 119862 (
10038171003817100381710038171198911003817100381710038171003817119871infin(R)10038171003817100381710038171205971199091198921003817100381710038171003817119867119904(R) +
10038171003817100381710038171198911003817100381710038171003817119867119904+1(R)10038171003817100381710038171198921003817100381710038171003817119871infin(R))
(32)
(c) For 1199041le 12 119904
2gt 12 119904
1+ 1199042gt 0
1003817100381710038171003817f11989210038171003817100381710038171198671199041 (R) le 119862
100381710038171003817100381711989110038171003817100381710038171198671199041 (R)
100381710038171003817100381711989210038171003817100381710038171198671199042 (R) (33)
where 1198621015840119904 are constants that are independent of 119891 and 119892
Proposition 9 (see [21]) Suppose that 119904 gt minus1198892 Let Vbe a vector field such that nablaV belongs to 119871
1([0 119879]119867
119904minus1) if
119904 gt 1 + 1198892 or to 1198711([0 119879]1198671198892 cap 119871infin) otherwise Suppose
also that 1198910
isin 119867119904 119865 isin 119871
1([0 119879]119867
119904) and that 119891 isin
119871infin([0 119879]119867
119904) cap 119862([0 119879] 119878
1015840) solves the 119889-dimensional linear
transport equations
(119879)
120597119905119891 + V sdot nabla119891 = 119865
1198911003816100381610038161003816119905=0
= 1198910
(34)
Then 119891 isin 119862([0 119879]119867119904) More precisely there exists a
constant119862 depending only on 119904119901 and119889 such that the followingstatements hold
(1) If 119904 = 1 + 1198892
10038171003817100381710038171198911003817100381710038171003817119867119904 le
10038171003817100381710038171198910
1003817100381710038171003817119867119904 + int
119905
0
119865 (120591)119867119904119889120591 + 119862int
119905
0
1198811015840(120591)
1003817100381710038171003817119891 (120591)
1003817100381710038171003817119867119904119889120591
(35)
or
10038171003817100381710038171198911003817100381710038171003817119867119904 le 119890119862119881(119905)
(10038171003817100381710038171198910
1003817100381710038171003817119867119904 + int
119905
0
119890minus119862119881(120591)
119865 (120591)119867119904119889120591) (36)
with 119881(119905) = int
119905
0nablaV(120591)
1198671198892cap119871infin119889120591 if 119904 lt 1 + 1198892 and 119881(119905) =
int
119905
0nablaV(120591)
119867119904minus1119889120591 else
Journal of Applied Mathematics 5
(2) If 119891 = V then for all 119904 gt 0 estimates (35) and (36) holdwith
119881 (119905) = int
119905
0
1003817100381710038171003817120597119909119906 (120591)
1003817100381710038171003817119871infin119889120591 (37)
Proposition 10 (see [21]) Let 0 lt 119904 lt 1 Suppose that 1198910isin
119867119904 119892 isin 119871
1([0 119879]119867
119904)V 120597119909V isin 119871
1([0 119879] 119871
infin) and that 119891 isin
119871infin([0 119879]119867
119904) cap 119862([0 119879] 119878
1015840) solves the 1-dimensional linear
transport equation
(119879)
120597119905119891 + V sdot nabla119891 = 119892
1198911003816100381610038161003816119905=0
= 1198910
(38)
Then 119891 isin 119862([0 119879]H119904) More precisely there exists a con-stant 119862 depending only on 119904 such that the following statementshold
10038171003817100381710038171198911003817100381710038171003817119867119904 le
10038171003817100381710038171198910
1003817100381710038171003817119867119904 + 119862int
119905
0
1003817100381710038171003817119892 (120591)
1003817100381710038171003817119867119904119889120591
+ 119862int
119905
0
1198811015840(120591)
1003817100381710038171003817119891 (120591)
1003817100381710038171003817119867119904119889120591
(39)
or
10038171003817100381710038171198911003817100381710038171003817119867119904 le 119890119862119881(119905)
(10038171003817100381710038171198910
1003817100381710038171003817119867119904 + 119862int
119905
0
1003817100381710038171003817119892 (120591)
1003817100381710038171003817119867119904119889120591) (40)
with 119881(119905) = int
119905
0(V(120591)
119871infin + 120597
119909V(120591)119871infin)119889120591
The above proposition was proved in [8] using Little-wood-Paley analysis for the transport equation and Moser-type estimates Using this result and performing the sameargument as in [17] we can obtain the following blow-upcriterion
Theorem 11 Let1198830= (1199060
1205880
) isin 119867119904times119867119904minus1 with 119904 ge 2 and119883 =
(119906
120588 ) be the corresponding solution to (3) Assume that 119879 gt 0 isthe maximal time of existence Then
119879 lt infin 997904rArr int
119879
0
1003817100381710038171003817120597119909119906 (120591)
1003817100381710038171003817119871infin119889120591 = infin (41)
Our next result describes the necessary and sufficientcondition for the blow-up of solutions to (3)
Theorem 12 Suppose that 120590 isin 119877 0 Let 1198830= (1199060
1205880
) isin 119867119904times
119867119904minus1 with 119904 ge 2 and let119879 be themaximal existence time of the
solution119883 = (119906
120588 ) to (3)with initial data1198830 Then the solutionblows up in finite time if and only if
lim inf119905rarr119879
minus
inf119909isin119878
120590119906119909(119905 119909) = minusinfin (42)
The approach one takes here is the method of charac-teristics Applying the following lemma we may carry outthe estimates along the characteristics 120593(119905 119909) which capturessup119909isin119878
119906119909(119905 119909) and inf
119909isin119878119906119909(119905 119909)
Lemma 13 (see [22]) Let119879 gt 0 and let V isin 1198621([0 119879]1198672(119877))Then for every 119905 isin [0 119879) there exists at least one point 120585(119905) isin 119877with
119898(119905) = inf119909isin119878
V119909(119905 119909) = V
119909(119905 120585 (119905)) (43)
and the function 119898(119905) is almost everywhere differentiable on(0 119879) with
119889119898 (119905)
119889119905
= V119905119909(119905 120585 (119905)) 119886119890 119900119899 (0 119879) (44)
Lemma 14 Let 1198830= (1199060
1205880
) isin 119867119904times 119867119904minus1 with 119904 ge 2 and let 119879
be the maximal existence time of the solution 119883 = (119906
120588 ) to (3)with initial data119883
0 Then one has the following
(1) 120590 = 0
sup119909isin119878
119906119909(119905 119909) le
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
+radic
10038171003817100381710038171205880
1003817100381710038171003817
2
119871infin(119878)+ 1198702
1(119879)
120590
+
1205822
1205902+
120582
120590
(120590 gt 0)
(45)
inf119909isin119878
119906119909(119905 119909) ge minus
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
minusradic1205822
1205902minus
1198702
2(119879)
120590
+
120582
120590
(120590 lt 0)
(46)
(2) 120590 = 0
sup119909isin119878
119906119909(119905 119909) le sup
119909isin119878
1199060119909
(119909)
+
1
2
(sup119909isin119878
1205882
0(119909) + 119870
2
1(119879))
119890120582119905minus 1
120582
(47)
inf119909isin119878
119906119909(119905 119909) ge inf
119909isin119878
1199060119909
(119909)
+
1
2
(inf119909isin119878
1205882
0(119909) minus 119870
2
2(119879))
119890120582119905minus 1
120582
(48)
The constants above are defined as follows
1198701(119879) = radic2119860 minus 120582 +
119860
2
1198640+
3119860 minus 2120582
2
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(49)
1198702(119879)
= radic2119860 minus 120582 +
119860 + 2
2
1198640+
3119860 minus 2120582
2
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(50)
Proof of Lemma 14 ByTheorem 2 and a simple density argu-ment we show that the desired results are valid when 119904 ge 3so we take 119904 = 3 in the proof
6 Journal of Applied Mathematics
Let 120590 gt 0 Using Lemma 13 and the fact that
sup119909isin119878
[V119909(119905 119909)] = minusinf
119909isin119878
[minusV119909(119905 119909)] (51)
We can consider119872(119905) and 120574(119905) as follows
119872(119905) = 119906119909(119905 120585 (119905)) = sup
119909isin119878
[119906119909(119905 119909)] 119905 isin [0 119879) (52)
Hence
119906119909119909(119905 120585 (119905)) = 0 ae on 119905 isin [0 119879) (53)
Take the trajectory 120593(119905 119909) defined in (12) Then we knowthat 120593(119905 sdot) 119877 rarr 119877 is a diffeomorphism for every 119905 isin [0 119879)Therefore there exists 119909
0(119905) isin 119877 such that
120593 (119905 1199090(119905)) = 120585 (119905) 119905 isin [0 119879) (54)
Now let
120574 (119905) = 120588 (119905 120593 (119905 1199090)) 119905 isin [0 119879) (55)
Therefore along the trajectory120593(119905 1199090) (4) and the second
equation of (3) become
1198721015840(119905) = minus
120590
2
1198722(119905) + 120582119872 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
1205741015840(119905) = minus120574119872 ae 119905 isin [0 119879)
(56)
where the notation denotes the derivative with respect to 119905
and 119891 represents the function
119891 = minus119860119906 minus 120582120597minus1
119909119906 + 119892 (119905)
= minus119860119906 minus 120582120597minus1
119909119906 minus int
119878
(
120590
2
1199062
119909+
1
2
1205882minus 119860119906)119889119909
(57)
We first compute the upper and lower bounds for 119891 forlater use in getting the blow-up result as follows
119891 = minus119860119906 minus 120582int
119909
0
119906 119889119909 minus
120590
2
int
119878
1199062
119909119889119909 minus
1
2
int
119878
1205882119889119909 + 119860int
119878
119906 119889119909
le minus119860119906 minus 120582int
119909
0
119906 119889119909 + 119860int
119878
119906 119889119909 le
119860
2
(1 + 1199062)
+
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
(58)
Since 1199062 le (12) int119878(1199062+ 1199062
119909)119889119909 (17) we obtain the upper
bound for 119891
119891 le
119860
2
(1 +
1
2
int
119878
(1199062+ 1199062
119909) 119889119909) +
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le 119860 minus
120582
2
+
119860
4
int
119878
(1205882+ 1199062
119909) 119889119909 +
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860
4
1198640+
3119860 minus 2120582
4
[1198901198622119905(int
119878
1199062
0(119909) 119889119909 + 1)]
le 119860 minus
120582
2
+
119860
4
1198640+
3119860 minus 2120582
4
times [1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)] =
1
2
1198702
1(119879)
(59)
Now we turn to the lower bound of 119891 Using previous argu-ments we get
minus119891 = 119860119906 + 120582int
119909
0
119906 119889119909 +
120590
2
int
119878
1199062
119909119889119909 +
1
2
int
119878
1205882119889119909 minus 119860int
119878
119906 119889119909
le 119860
1 + 1199062
2
+
max (|120590| 1)2
int
119878
(1205882+ 1199062
119909) 119889119909
+
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le 119860 minus
120582
2
+
119860 + 2max (|120590| 1)4
int
119878
(1205882+ 1199062
119909) 119889119909
+
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860 + 2max (|120590| 1)4
1198640
+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(60)
When 120590 lt 0 we have a finer estimate
minus119891 le 119860 minus
120582
2
+
119860 + 2
4
int
119878
(1205882+ 1199062
119909) 119889119909 +
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860 + 2
4
1198640+
3119860 minus 2120582
4
times [1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)] =
1
2
1198702
2(119879)
(61)
Combining (59) and (60) we obtain
10038161003816100381610038161198911003816100381610038161003816le 119860 minus
120582
2
+
119860 + 2max (|120590| 1)4
1198640
+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)] =
1
2
1198702
3(119879)
(62)
Since 119904 ge 3 we have 119906 isin 11986210(119878) Therefore
sup119909isin119878
119906119909(119905 119909) ge 0 inf
119909isin119878
119906119909(119905 119909) le 0 119905 isin [0 119879) (63)
Journal of Applied Mathematics 7
Hence119872(119905) gt 0 for 119905 isin [0 119879) From the second equationof (55) we obtain
120574 (119905) = 120574 (0) 119890minusint119905
0
119872(120591)119889120591 (64)
Hence1003816100381610038161003816120588 (119905 120593 (119905 119909
0))1003816100381610038161003816=1003816100381610038161003816120574 (119905)
1003816100381610038161003816le1003816100381610038161003816120574 (0)
1003816100381610038161003816le10038171003817100381710038171205880
1003817100381710038171003817119871infin(119878) (65)
For any given 119909 isin 119878 define
1199011(119905) = 119872 (119905) minus
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
minusradic
10038171003817100381710038171205880
1003817100381710038171003817
2
119871infin(119878)+ 1198702
1(119879)
120590
+
1205822
1205902minus
120582
120590
(120590 gt 0)
(66)
Observing that 1199011(119905) is a 1198621-differentiable function on
[0 119879) and satisfies
1199011(0) = 119872 (0) minus
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
minusradic
10038171003817100381710038171205880
1003817100381710038171003817
2
119871infin(119878)+ 1198702
1(119879)
120590
+
1205822
1205902minus
120582
120590
le 119872(0) minus10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
(67)
We now claim that 1199011(119905) le 0 119905 isin [0 119879)
Assume the contrary that there is 1199050isin [0 119879) such that
1199011(1199050) gt 0
Let 1199051= max119905 lt 119905
0 1199011(119905) = 0 Then 119901
1(1199051) = 0 and
1199011015840
1(1199051) ge 0 or equivalently
119872(1199051) =
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)+radic
10038171003817100381710038171205880
1003817100381710038171003817
2
119871infin(119878)+ 1198702
1(119879)
120590
+
1205822
1205902+
120582
120590
(68)
and1198721015840(1199051) ge 0 ae 119905 isin [0 119879) On the other hand we have
1198721015840(1199051) = minus
120590
2
1198722(1199051) + 120582119872(119905
1) +
1
2
1205742(1199051)
+ 119891 (1199051 120593 (1199051 1199090)) ae 119905 isin [0 119879)
le minus
120590
2
(119872(1199051) minus
120582
120590
)
2
+
1
2
10038171003817100381710038171205880
1003817100381710038171003817
2
119871infin(119878)+
1205822
2120590
+
1198702
1(119879)
2
lt 0
(69)
which is a contradictionTherefore1199011(119905) le 0 for all 119905 isin [0 119879)
Since 119909 is arbitrarily chosen we obtain (45)To derive (46) in the case of 120590 lt 0 we consider (119905) and
120574(119905) as in Lemma 13
(119905) = 119906119909(119905 120577 (119905)) = inf
119909isin119878
[119906119909(119905 119909)] 119905 isin [0 119879) (70)
Hence
119906119909119909(119905 120577 (119905)) = 0 ae 119905 isin [0 119879) (71)
Using previous arguments we take the characteristic 120593(119905119909) defined in (13) and choose 119909
1(119905) isin 119877 such that
120593 (119905 1199091(119905)) = 120577 (119905) (72)
Let
120574 (119905) = 120588 (119905 120593 (119905 119909)) 119905 isin [0 119879) (73)
Hence along the trajectory 1015840(119905) = 120582(119905) +(12)1205742(119905) +
119891(119905 120593(119905 1199090)) ge 120582(0)+(12)120574
2(0) +(12)119870
2
2(119879) (4) and the
second equation of (3) become
1015840(119905) = minus
120590
2
2(119905) + 120582 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
1205741015840(119905) = minus120574 ae 119905 isin [0 119879)
(74)
Define
1199012(119905) = (119905) +
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
+radic1205822
1205902minus
1198702
2(119879)
120590
minus
120582
120590
(120590 lt 0)
(75)
For any given 119909 isin 119878 Note that 1199012(119905) is also 119862
1-dif-ferentiable function on [0 119879) and satisfies
1199012(0) = (0) +
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
+radic1205822
1205902minus
1198702
2(119879)
120590
minus
120582
120590
ge (0) +10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
ge 0
(76)
We now claim that 1199012(119905) ge 0 for any 119905 isin [0 119879)
Suppose not then there is isin [0 119879) such that 1199012(1199050) lt 0
Define
1199052= max 119905 lt 119905
0 1199012(119905) = 0 (77)
Then 1199012(1199052) = 0 and 1199011015840
2(1199052) lt 0 or equivalently
(1199052) = minus
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)minusradic1205822
1205902minus
1198702
2(119879)
120590
+
120582
120590
(78)
and 1015840(1199052) le 0 ae 119905 isin [0 119879) However we have
1015840(1199052) = minus
120590
2
2(1199052) + 120582 (119905
2) +
1
2
1205742(1199052)
+ 119891 (1199052 120593 (1199052 1199090)) ae 119905 isin [0 119879)
ge minus
120590
2
(minus10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)minusradic1205822
1205902minus
1198702
2(119879)
120590
)
2
+
1205822
2120590
minus
1
2
1198702
2(119879) gt 0
(79)
8 Journal of Applied Mathematics
Therefore 1199012(119905) ge 0 for any 119905 isin [0 119879) Since 119909 is chosen
arbitrarily we obtain (46)Let 120590 = 0 Using previous arguments (56) becomes
1198721015840(119905) = minus
120590
2
1198722(119905) + 120582119872 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
1205741015840(119905)
= minus120574119872 ae 119905 isin [0T) (80)
where the notation denotes the derivative with respect to 119905
and 119891 represents the function
119891 = minus119860119906 minus 120582120597minus1
119909119906 + 119892 (119905)
= minus119860119906 minus 120582120597minus1
119909119906 minus int
119878
(
1
2
1205882minus 119860119906)119889119909
(81)
We first compute the upper and lower bounds for 119891 forlater use in getting the blow-up result
119891 = minus119860119906 minus 120582int
119909
0
119906 119889119909 minus
1
2
int
119878
1205882119889119909 + 119860int
119878
119906 119889119909
le minus119860119906 minus 120582int
119909
0
119906 119889119909 + 119860int
119878
119906 119889119909 le
119860
2
(1 + 1199062)
+
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le
119860
2
(1 +
1
2
int
119878
(1199062+ 1199062
119909) 119889119909) +
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le 119860 minus
120582
2
+
119860
4
int
119878
(1205882+ 1199062
119909) 119889119909 +
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860
4
1198640+
3119860 minus 2120582
4
[1198901198622119905(int
119878
1199062
0(119909) 119889119909 + 1)]
le 119860 minus
120582
2
+
119860
4
1198640+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(82)
Now we turn to the lower bound of 119891
minus119891 = 119860119906 + 120582int
119909
0
119906 119889119909 +
1
2
int
119878
1205882119889119909 minus 119860int
119878
119906 119889119909
le 119860
1 + 1199062
2
+
1
2
int
119878
1205882119889119909 +
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le 119860 minus
120582
2
+
119860 + 2
4
int
119878
(1205882+ 1199062
119909) 119889119909 +
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860 + 2
4
1198640+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(83)
Combining (82) and (83) we obtain
10038161003816100381610038161198911003816100381610038161003816le 119860 minus
120582
2
+
119860 + 2
4
1198640+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(84)
We know 119872(119905) gt 0 for 119905 isin [0 119879) From the secondequation of (81) we obtain that
120574 (119905) = 120574 (0) 119890minusint119905
0
119872(120591)119889120591 (85)
Hence1003816100381610038161003816120588 (119905 120593 (119905 119909
0))1003816100381610038161003816=1003816100381610038161003816120574 (119905)
1003816100381610038161003816le1003816100381610038161003816120574 (0)
1003816100381610038161003816 (86)
Therefore we have
1198721015840(119905) = 120582119872 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
le 120582119872 (0) +
1
2
1205742(0) +
1
2
1198702
1(119879)
(87)
1198721015840(119905) minus 120582119872 (119905) le
1
2
(sup119909isin119878
1205882
0(119909) + 119870
2
1(119879)) ae 119905 isin [0 119879)
(88)
Integrating (88) on [0 119905] we prove (47) as follows
119872(119905) le sup119909isin119878
1199060119909
(119909) +
1
2
(sup119909isin119878
1205882
0(119909) + 119870
2
1(119879))
119890120582119905minus 1
120582
(89)
To obtain a lower bound for inf119909isin119878
119906119909(119905 119909) we use the
same argumentSince 120590 = 0 (80) becomes
1015840(119905) = 120582 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
1))
1205741015840(119905) = minus120574 ae 119905 isin [0 119879)
(90)
Because of (119905) lt 0 we get from the second equation of(90) that
120574 (119905) = 120574 (0) 119890minusint119905
0
119872(120591)119889120591 (91)
This means that1003816100381610038161003816120588 (119905 120593 (119905 119909
1))1003816100381610038161003816=1003816100381610038161003816120574 (119905)
1003816100381610038161003816ge1003816100381610038161003816120574 (0)
1003816100381610038161003816 (92)
Then
1015840(119905) = 120582 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
ge 120582 (0) +
1
2
1205742(0) +
1
2
1198702
2(119879)
(93)
1015840(119905) minus 120582 (119905) ge
1
2
(inf119909isin119878
1205882
0(119909) minus 119870
2
2(119879)) ae 119905 isin [0 119879)
(94)
Integrating (94) on [0 119905] we prove (48) This completesthe proof of Lemma 14
Lemma 15 Suppose that 120590 isin 119877 0 Suppose 1198830= (1199060
1205880
) isin
119867119904times119867119904minus1 with 119904 ge 2 and let 119879 be the maximal existence time
Journal of Applied Mathematics 9
of the solution 119883 = (119906
120588 ) to (3) with initial data 1198830 Then we
have
120588 (119905 120593 (119905 119909)) 120593119909(119905 119909) = 120588
0(119909) (119905 119909) isin [0 119879] times 119878 (95)
Moreover if there exists119872 gt 0 such that
inf(119905119909)isin[0119879]times119878
120590119906119909(119905 119909) ge minus119872 (119905 119909) isin [0 119879] times 119878 (96)
Then1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin(119878)
=1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin(119878)
le 119890119872119879120590 1003817
1003817100381710038171205880
1003817100381710038171003817119871infin(119878)
(120590 gt 0)
(97)
1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin(119878)
=1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin(119878)
le 119890119873119879120590 1003817
1003817100381710038171205880
1003817100381710038171003817119871infin(119878)
(120590 lt 0)
(98)
where119873 = 1199060119909119871infin(119878)+radic12058221205902minus 1198702
2(119879)120590 minus 120582120590 and119870
2(119879)
is given in (50)
Proof Differentiating the left hand side of (95) with respectto 119905 in view of the relations (12) and (3) we obtain
119889
119889119905
120588 (119905 120593 (119905 119909)) 120593119909(119905 119909)
= [120588119905(119905 120593) + 120588
119909(119905 120593) 120593
119905(119905 119909)]
times 120593119909(119905 119909) + 120588 (119905 120593) 120593
119909119905(119905 119909)
= [120588119905(119905 120593) + 120588
119909(119905 120593) 120593
119905(119905 119909)] 120593
119909(119905 119909)
+ 120588 (119905 120593) 119906119909(119905 120593) 120593
119909(119905 119909)
= [120588119905(119905 120593) + 120588
119909(119905 120593) 120593
119905(119905 119909) + 120588 (119905 120593) 119906
119909(119905 120593)]
times 120593119909(119905 119909) = 0
(99)
This completes the proof of (95) In view of the assump-tion (96) and 120590 gt 0 we obtain 119906(119905 119909) ge minus(119872120590) (119905 119909) isin
[0 119879] times 119878By Lemma 3 and (95) we have
1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin =
1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin =
1003817100381710038171003817100381710038171003817
119890minusint119905
0
119906119909(120591sdot)119889120591
1205880(sdot)
1003817100381710038171003817100381710038171003817119871infin
le 1198901198721198791205901003817
1003817100381710038171205880(sdot)1003817100381710038171003817119871infin
(100)
To obtain (98) we use a similar argument as before Using(13) and the lower bound for 119906
119909(119905 119909) in (46) it follows that
1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin =
1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin =
1003817100381710038171003817100381710038171003817
119890minusint119905
0
119906119909(120591sdot)119889120591
1205880(sdot)
1003817100381710038171003817100381710038171003817119871infin
le 1198901198731198791205901003817
1003817100381710038171205880(sdot)1003817100381710038171003817119871infin
(101)
which proves (98) This completes the proof of Lemma 15
Proof of Theorem 12 Suppose that119879 lt infin and that (42) is notvalid Then there is some positive number119872 gt 0 such that
120590119906119909(119905 119909) ge minus119872 (119905 119909) isin [0 119879] times 119878 (102)
It now follows from Lemma 14 that |119906119909(119905 119909)| le 119862 where
119862 = 119862(119860119872 120590 1198640 120582 119906
0 119879) Therefore Theorem 11 implies
that the maximal existence time 119879 = infin which contradictswith the assumption that 119879 lt infin
Conversely the Sobolev embedding theorem 119867119904(119878) rarr
119871infin(119878) with 119904 gt 12 implies that if (70) holds the corre-
sponding solution blows up in finite time which completesthe proof of Theorem 12
Acknowledgments
The authors would like to thank the referee for commentsand suggestions This work is supported by the NationalNature Science Foundation of China (no 11171135) theNature Science Foundation of the Jiangsu Higher EducationInstitutions of China (no 09KJB110003) and the high-leveltalented person special subsidizes of Jiangsu University (no05JDG047)
References
[1] J K Hunter and R Saxton ldquoDynamics of director fieldsrdquo SIAMJournal on Applied Mathematics vol 51 no 6 pp 1498ndash15211991
[2] Z Yin ldquoOn the structure of solutions to the periodic Hunter-Saxton equationrdquo SIAM Journal on Mathematical Analysis vol36 no 1 pp 272ndash283 2004
[3] H-H Dai and M Pavlov ldquoTransformations for the Camassa-Holm equation its high-frequency limit and the Sinh-Gordonequationrdquo Journal of the Physical Society of Japan vol 67 no 11pp 3655ndash3657 1998
[4] J K Hunter and Y X Zheng ldquoOn a completely integrablenonlinear hyperbolic variational equationrdquo Physica D vol 79no 2ndash4 pp 361ndash386 1994
[5] R Camassa and D D Holm ldquoAn integrable shallow waterequation with peaked solitonsrdquo Physical Review Letters vol 71no 11 pp 1661ndash1664 1993
[6] R S Johnson ldquoCamassa-Holm Korteweg-de Vries and relatedmodels for water wavesrdquo Journal of Fluid Mechanics vol 455pp 63ndash82 2002
[7] A Constantin and B Kolev ldquoOn the geometric approach to themotion of inertialmechanical systemsrdquo Journal of Physics A vol35 no 32 pp R51ndashR79 2002
[8] P J Olver and P Rosenau ldquoTri-Hamiltonian duality betweensolitons and solitary-wave solutions having compact supportrdquoPhysical Review E vol 53 no 2 pp 1900ndash1906 1996
[9] R Beals D H Sattinger and J Szmigielski ldquoInverse scatteringsolutions of the Hunter-Saxton equationrdquo Applicable Analysisvol 78 no 3-4 pp 255ndash269 2001
[10] A Bressan and A Constantin ldquoGlobal solutions of the Hunter-Saxton equationrdquo SIAM Journal on Mathematical Analysis vol94 pp 68ndash92 2010
[11] A Bressan H Holden and X Raynaud ldquoLipschitz metric forthe Hunter-Saxton equationrdquo Journal de Mathematiques Pureset Appliquees vol 94 no 1 pp 68ndash92 2010
10 Journal of Applied Mathematics
[12] J K Hunter and Y X Zheng ldquoOn a nonlinear hyperbolicvariational equation I Global existence of weak solutionsrdquoArchive for Rational Mechanics and Analysis vol 129 no 4 pp305ndash353 1995
[13] J Lenells ldquoThe Hunter-Saxton equation describes the geodesicflow on a sphererdquo Journal of Geometry and Physics vol 57 no10 pp 2049ndash2064 2007
[14] J B Li and Y S Li ldquoBifurcations of travelling wave solutions fora two-component Camassa-Holm equationrdquoActaMathematicaSinica (English Series) vol 24 no 8 pp 1319ndash1330 2008
[15] W Rui and Y Long ldquoIntegral bifurcation method together witha translation-dilation transformation for solving an integrable2-component Camassa-Holm shallow water systemrdquo Journal ofApplied Mathematics vol 2012 Article ID 736765 21 pages2012
[16] H Wu and M Wunsch ldquoGlobal existence for the generalizedtwo-component Hunter-Saxton systemrdquo Journal of Mathemati-cal Fluid Mechanics vol 14 no 3 pp 455ndash469 2012
[17] BMoon andY Liu ldquoWave breaking and global existence for thegeneralized periodic two-component Hunter-Saxton systemrdquoJournal of Differential Equations vol 253 no 1 pp 319ndash3552012
[18] S Wu and Z Yin ldquoBlow-up blow-up rate and decay of thesolution of the weakly dissipative Camassa-Holm equationrdquoJournal ofMathematical Physics vol 47 no 1 Article ID 0135042006
[19] S Wu J Escher and Z Yin ldquoGlobal existence and blow-up phenomena for a weakly dissipative Degasperis-Procesiequationrdquo Discrete and Continuous Dynamical Systems B vol12 no 3 pp 633ndash645 2009
[20] T Kato ldquoQuasi-linear equations of evolution with applicationsto partial differential equationsrdquo in Spectral Theory and Differ-ential Equations vol 448 of Lecture Notes in Mathematics pp25ndash70 Springer Berlin Germany 1975
[21] G Gui and Y Liu ldquoOn the global existence and wave-breakingcriteria for the two-component Camassa-Holm systemrdquo Journalof Functional Analysis vol 258 no 12 pp 4251ndash4278 2010
[22] A Constantin and J Escher ldquoWave breaking for nonlinearnonlocal shallow water equationsrdquo Acta Mathematica vol 181no 2 pp 229ndash243 1998
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 3
Now discuss the initial value problem for the Lagrangian flowmap as follows
120597120593
120597119905
= 119906 (119905 120593 (119905 119909)) 119905 isin [0 119879)
120593 (0 119909) = 119909 119909 isin 119877
(12)
where 119906 is the first component of the solution 119883 to (3) Usingclassical results from ordinary differential equations one canacquire the following result on 120593which is of vital importance inthe proof of the blow-up scenarios
Lemma 3 (see [17]) Let 119906 isin 119862([0 119879)119867119904) cap 1198621([0 119879)119867119904minus1)119904 ge 2 Then initial value problem (12) admits a unique solu-tion 120593 isin 119862
1([0 119879) times 119877 119877) Moreover 120593(119905 sdot)
119905isin[0119879)is increas-
ing diffeomorphism of 119877 with
120593119909(119905 119909) = 119890
int119879
0
119906119909(120591120593(120591119909))119889120591
gt 0 (119905 119909) isin [0 119879) times 119877 (13)
Remark 4 Since 120593(119905 sdot) 119877 rarr 119877 is a diffeomorphism ofthe linear for every 119905 isin [0 119879) the 119871infin-norm of any functionV(119905 sdot) isin 119871
infin 119905 isin [0 119879) is preserved under the family ofdiffeomorphisms 120593(119905 sdot) with 119905 isin [0 119879) that is
V (119905 sdot)119871infin(119878)
=1003817100381710038171003817V (119905 120593 (119905 sdot))100381710038171003817
1003817119871infin(119878) 119905 isin [0 119879) (14)
Similarly we have
inf119909isin119878
V (119905 119909) = inf119909isin119878
V (119905 120593 (119905 119909)) 119905 isin [0 119879)
sup119909isin119878
V (119905 119909) = sup119909isin119878
V (119905 120593 (119905 119909)) 119905 isin [0 119879)
(15)
Lemma 5 Let 1198830= (1199060
1205880
) isin 119867119904times 119867119904minus1 119904 ge 2 and let 119879 be
the maximal existence time of the solution119883 = (119906
120588 ) to (3) withinitial data 119883
0 Then for all 119905 isin [0 119879) we have the following
results
int
119878
120588 (119905 119909) 119889119909 = int
119878
1205880(119905) 119889119909 (16)
int
119878
1199062
119909(119905 119909) + 120588
2(119905 119909) 119889119909 le int
119878
1199062
0119909(119909) + 120588
2
0(119909) 119889119909 ≜ 119864
0
(17)
Proof On the one hand integrating the second equation in(3) by parts and using the periodicity of 119906 and 120588 we acquire
119889
119889119905
int
119878
120588 119889119909 = minusint
119878
(119906120588)119909119889119909 = 0 (18)
On the other hand multiplying (4) by 119906119909and integrating
by parts considering the periodicity of 119906 we obtain
119889
119889119905
int
119878
1199062
119909119889119909 = minus2int
119878
119906120588120588119909119889119909 + 2120582int
119878
1199062119889119909 + 2120582int
119878
1199062
119909119889119909
(19)
Multiplying the second equation in (3) by 120588 and integrat-ing by parts we have
119889
119889119905
int
119878
1205882119889119909 = 2int
119878
119906120588120588119909119889119909 (20)
Adding the above two equations we get
119889
119889119905
int
119878
1199062
119909+ 1205882119889119909 = 2120582119906
2
1198672 120582 lt 0 (21)
We acquire
int
119878
1199062
119909(119905 119909) + 120588
2(119905 119909) 119889119909 le int
119878
1199062
0119909(119909) + 120588
2
0(119909) 119889119909 ≜ 119864
0
(22)
This completes the proof of Lemma 5
Lemma 6 Let 1198830= (1199060
1205880
) isin 119867119904times 119867119904minus1 119904 ge 2 and let 119879 be
the maximal existence time of the solution119883 = (119906
120588 ) to (3) withinitial data 119883
0 Then for all 119905 isin [0 119879) we have the following
results
int
119878
1199062(119905 119909) 119889119909 le 119890
1198622119905(int
119878
1199062
0(119909) 119889119909 + 1) (23)
where 1198621= max(|120590| minus 120582 1)119864
0+ sup
119905isin[0infin)|ℎ(119905)| gt 0 119862
2=
1198621+ 4119860 minus 2120582
Proof By computing directly we have
1003816100381610038161003816100381610038161003816
120597minus1
119909(
120590
2
1199062
119909+
1
2
1205882minus 119860119906 + 119892 + 120582119906
119909minus 120582120597minus1
119909119906) + ℎ (119905)
1003816100381610038161003816100381610038161003816
le int
119909
0
1003816100381610038161003816100381610038161003816
120590
2
1199062
119909+
1
2
1205882minus 119860119906 + 119892 + 120582119906
119909minus 120582120597minus1
119909119906
1003816100381610038161003816100381610038161003816
119889119909 + |ℎ (119905)|
le
1
2
max (|120590| minus 120582 1) 1198640+1003816100381610038161003816119892 (119905)
1003816100381610038161003816+ |ℎ (119905)|
+ (|119860| minus 120582) int
1
0
|119906| 119889119909 minus 120582
le max (|120590| minus 120582 1) 1198640+ |ℎ (119905)| + (2 |119860| minus 120582) int
119878
|119906| 119889119909 minus 120582
= 1198621+ (2 |119860| minus 120582) int
119878
|119906| 119889119909 minus 120582
(24)
where 1198621= max(|120590| minus 120582 1)119864
0+ sup119905isin[0infin)
|ℎ(119905)| gt 0 and
1003816100381610038161003816119892 (119905)
1003816100381610038161003816=
10038161003816100381610038161003816100381610038161003816
minus int
119878
(
120590
2
1199062
119909+
1
2
1205882minus 119860119906)119889119909
10038161003816100381610038161003816100381610038161003816
le
1
2
max (|120590| 1) 1198640+ 119860int
119878
|119906| 119889119909
(25)
4 Journal of Applied Mathematics
Multiplying (6) by 119906 and integrating with respect to 119909using the periodicity of 119906 and (24) we obtain
1
2
119889
119889119905
int
119878
1199062(119905 119909) 119889119909
= int
119878
119906119906119905119889119909
= minus120590int
119878
1199061199091199062119889119909
+ int
119878
119906 [120597minus1
119909(
120590
2
1199062
119909+
1
2
1205882minus 119860119906 + 120582119906
119909
minus 120582120597minus1
119909119906 + 119892) +ℎ (119905) ] 119889119909
= int
119878
119906 [120597minus1
119909(
120590
2
1199062
119909+
1
2
1205882minus 119860119906 + 120582119906
119909
minus120582120597minus1
119909119906 + 119892) + ℎ (119905) ] 119889119909
le (1198621+ (2 |119860| minus 120582) int
119878
|119906| 119889119909 minus 120582)int
119878
|119906| 119889119909
le 1198621int
119878
|119906| 119889119909 + (2 |119860| minus 120582) (int
119878
|119906| 119889119909)
2
minus 120582int
119878
1199062119889119909
le (
1198621
2
+ (2 |119860| minus 2120582))int
119878
1199062119889119909 +
1198621minus 120582
2
=
1198622
2
int
119878
1199062119889119909 +
1198621minus 120582
2
(26)
where 1198622= 1198621+ 4119860 minus 2120582 note that 119862
2gt 1198621
By Gronwallrsquos inequality we get
int
119878
1199062(119905 119909) 119889119909 le 119890
1198622119905(int
119878
1199062
0(119909) 119889119909 +
1198621minus 120582
1198622
) minus
1198621minus 120582
1198622
le 1198901198622119905(int
119878
1199062
0(119909) 119889119909 + 1)
(27)
This completes the proof of Lemma 6
Lemma 7 Assume that 1199060isin 119867119904(119878) 119904 ge 2 119906
0= 0 and that
the corresponding solution 119906(119905 119909) of (3) has a zero point forany time 119905 ge 0 Then for all 119905 isin [0 119879) we have
int
119878
1199062(119905 119909) 119889119909 le int
119878
1199062
119909(119905 119909) 119889119909 le 119864
0 (28)
Proof By assumption there is 1199090isin [0 1] such that 119906(119905 119909
0) =
0 for each 119905 isin [0 119879)Then for 119909 isin 119878 by holder equality we have
1199062(119905 119909) = (int
119909
1199090
119906119909119889119909)
2
le (119909 minus 1199090) int
119909
1199090
1199062
119909119889119909
119909 isin [1199090 1199090+
1
2
]
(29)
This implies sup119909isin119878
1199062(119905 119909) le (12) int
1198781199062
119909119889119909
int
119878
1199062(119905 119909) 119889119909 le sup
119909isin119878
1199062(119905 119909) le
1
2
int
119878
1199062
119909119889119909
le int
119878
1199062
119909+ 1205882119889119909 le 119864
0
(30)
3 Wave-Breaking Criteria
In this section by using transport equation theory we obtainthe wave-breaking criteria for solutions to (3) We first recallthe following propositions
Proposition 8 (1D Moser-type estimates) The following esti-mates hold
(a) For 119904 ge 010038171003817100381710038171198911198921003817100381710038171003817119867119904(R) le 119862 (
10038171003817100381710038171198911003817100381710038171003817119871infin(R)10038171003817100381710038171198921003817100381710038171003817119867119904(R) +
10038171003817100381710038171198911003817100381710038171003817119867119904(R)10038171003817100381710038171198921003817100381710038171003817119871infin(R)) (31)
(b) For 119904 gt 010038171003817100381710038171198911205971199091198921003817100381710038171003817119867119904(R) le 119862 (
10038171003817100381710038171198911003817100381710038171003817119871infin(R)10038171003817100381710038171205971199091198921003817100381710038171003817119867119904(R) +
10038171003817100381710038171198911003817100381710038171003817119867119904+1(R)10038171003817100381710038171198921003817100381710038171003817119871infin(R))
(32)
(c) For 1199041le 12 119904
2gt 12 119904
1+ 1199042gt 0
1003817100381710038171003817f11989210038171003817100381710038171198671199041 (R) le 119862
100381710038171003817100381711989110038171003817100381710038171198671199041 (R)
100381710038171003817100381711989210038171003817100381710038171198671199042 (R) (33)
where 1198621015840119904 are constants that are independent of 119891 and 119892
Proposition 9 (see [21]) Suppose that 119904 gt minus1198892 Let Vbe a vector field such that nablaV belongs to 119871
1([0 119879]119867
119904minus1) if
119904 gt 1 + 1198892 or to 1198711([0 119879]1198671198892 cap 119871infin) otherwise Suppose
also that 1198910
isin 119867119904 119865 isin 119871
1([0 119879]119867
119904) and that 119891 isin
119871infin([0 119879]119867
119904) cap 119862([0 119879] 119878
1015840) solves the 119889-dimensional linear
transport equations
(119879)
120597119905119891 + V sdot nabla119891 = 119865
1198911003816100381610038161003816119905=0
= 1198910
(34)
Then 119891 isin 119862([0 119879]119867119904) More precisely there exists a
constant119862 depending only on 119904119901 and119889 such that the followingstatements hold
(1) If 119904 = 1 + 1198892
10038171003817100381710038171198911003817100381710038171003817119867119904 le
10038171003817100381710038171198910
1003817100381710038171003817119867119904 + int
119905
0
119865 (120591)119867119904119889120591 + 119862int
119905
0
1198811015840(120591)
1003817100381710038171003817119891 (120591)
1003817100381710038171003817119867119904119889120591
(35)
or
10038171003817100381710038171198911003817100381710038171003817119867119904 le 119890119862119881(119905)
(10038171003817100381710038171198910
1003817100381710038171003817119867119904 + int
119905
0
119890minus119862119881(120591)
119865 (120591)119867119904119889120591) (36)
with 119881(119905) = int
119905
0nablaV(120591)
1198671198892cap119871infin119889120591 if 119904 lt 1 + 1198892 and 119881(119905) =
int
119905
0nablaV(120591)
119867119904minus1119889120591 else
Journal of Applied Mathematics 5
(2) If 119891 = V then for all 119904 gt 0 estimates (35) and (36) holdwith
119881 (119905) = int
119905
0
1003817100381710038171003817120597119909119906 (120591)
1003817100381710038171003817119871infin119889120591 (37)
Proposition 10 (see [21]) Let 0 lt 119904 lt 1 Suppose that 1198910isin
119867119904 119892 isin 119871
1([0 119879]119867
119904)V 120597119909V isin 119871
1([0 119879] 119871
infin) and that 119891 isin
119871infin([0 119879]119867
119904) cap 119862([0 119879] 119878
1015840) solves the 1-dimensional linear
transport equation
(119879)
120597119905119891 + V sdot nabla119891 = 119892
1198911003816100381610038161003816119905=0
= 1198910
(38)
Then 119891 isin 119862([0 119879]H119904) More precisely there exists a con-stant 119862 depending only on 119904 such that the following statementshold
10038171003817100381710038171198911003817100381710038171003817119867119904 le
10038171003817100381710038171198910
1003817100381710038171003817119867119904 + 119862int
119905
0
1003817100381710038171003817119892 (120591)
1003817100381710038171003817119867119904119889120591
+ 119862int
119905
0
1198811015840(120591)
1003817100381710038171003817119891 (120591)
1003817100381710038171003817119867119904119889120591
(39)
or
10038171003817100381710038171198911003817100381710038171003817119867119904 le 119890119862119881(119905)
(10038171003817100381710038171198910
1003817100381710038171003817119867119904 + 119862int
119905
0
1003817100381710038171003817119892 (120591)
1003817100381710038171003817119867119904119889120591) (40)
with 119881(119905) = int
119905
0(V(120591)
119871infin + 120597
119909V(120591)119871infin)119889120591
The above proposition was proved in [8] using Little-wood-Paley analysis for the transport equation and Moser-type estimates Using this result and performing the sameargument as in [17] we can obtain the following blow-upcriterion
Theorem 11 Let1198830= (1199060
1205880
) isin 119867119904times119867119904minus1 with 119904 ge 2 and119883 =
(119906
120588 ) be the corresponding solution to (3) Assume that 119879 gt 0 isthe maximal time of existence Then
119879 lt infin 997904rArr int
119879
0
1003817100381710038171003817120597119909119906 (120591)
1003817100381710038171003817119871infin119889120591 = infin (41)
Our next result describes the necessary and sufficientcondition for the blow-up of solutions to (3)
Theorem 12 Suppose that 120590 isin 119877 0 Let 1198830= (1199060
1205880
) isin 119867119904times
119867119904minus1 with 119904 ge 2 and let119879 be themaximal existence time of the
solution119883 = (119906
120588 ) to (3)with initial data1198830 Then the solutionblows up in finite time if and only if
lim inf119905rarr119879
minus
inf119909isin119878
120590119906119909(119905 119909) = minusinfin (42)
The approach one takes here is the method of charac-teristics Applying the following lemma we may carry outthe estimates along the characteristics 120593(119905 119909) which capturessup119909isin119878
119906119909(119905 119909) and inf
119909isin119878119906119909(119905 119909)
Lemma 13 (see [22]) Let119879 gt 0 and let V isin 1198621([0 119879]1198672(119877))Then for every 119905 isin [0 119879) there exists at least one point 120585(119905) isin 119877with
119898(119905) = inf119909isin119878
V119909(119905 119909) = V
119909(119905 120585 (119905)) (43)
and the function 119898(119905) is almost everywhere differentiable on(0 119879) with
119889119898 (119905)
119889119905
= V119905119909(119905 120585 (119905)) 119886119890 119900119899 (0 119879) (44)
Lemma 14 Let 1198830= (1199060
1205880
) isin 119867119904times 119867119904minus1 with 119904 ge 2 and let 119879
be the maximal existence time of the solution 119883 = (119906
120588 ) to (3)with initial data119883
0 Then one has the following
(1) 120590 = 0
sup119909isin119878
119906119909(119905 119909) le
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
+radic
10038171003817100381710038171205880
1003817100381710038171003817
2
119871infin(119878)+ 1198702
1(119879)
120590
+
1205822
1205902+
120582
120590
(120590 gt 0)
(45)
inf119909isin119878
119906119909(119905 119909) ge minus
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
minusradic1205822
1205902minus
1198702
2(119879)
120590
+
120582
120590
(120590 lt 0)
(46)
(2) 120590 = 0
sup119909isin119878
119906119909(119905 119909) le sup
119909isin119878
1199060119909
(119909)
+
1
2
(sup119909isin119878
1205882
0(119909) + 119870
2
1(119879))
119890120582119905minus 1
120582
(47)
inf119909isin119878
119906119909(119905 119909) ge inf
119909isin119878
1199060119909
(119909)
+
1
2
(inf119909isin119878
1205882
0(119909) minus 119870
2
2(119879))
119890120582119905minus 1
120582
(48)
The constants above are defined as follows
1198701(119879) = radic2119860 minus 120582 +
119860
2
1198640+
3119860 minus 2120582
2
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(49)
1198702(119879)
= radic2119860 minus 120582 +
119860 + 2
2
1198640+
3119860 minus 2120582
2
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(50)
Proof of Lemma 14 ByTheorem 2 and a simple density argu-ment we show that the desired results are valid when 119904 ge 3so we take 119904 = 3 in the proof
6 Journal of Applied Mathematics
Let 120590 gt 0 Using Lemma 13 and the fact that
sup119909isin119878
[V119909(119905 119909)] = minusinf
119909isin119878
[minusV119909(119905 119909)] (51)
We can consider119872(119905) and 120574(119905) as follows
119872(119905) = 119906119909(119905 120585 (119905)) = sup
119909isin119878
[119906119909(119905 119909)] 119905 isin [0 119879) (52)
Hence
119906119909119909(119905 120585 (119905)) = 0 ae on 119905 isin [0 119879) (53)
Take the trajectory 120593(119905 119909) defined in (12) Then we knowthat 120593(119905 sdot) 119877 rarr 119877 is a diffeomorphism for every 119905 isin [0 119879)Therefore there exists 119909
0(119905) isin 119877 such that
120593 (119905 1199090(119905)) = 120585 (119905) 119905 isin [0 119879) (54)
Now let
120574 (119905) = 120588 (119905 120593 (119905 1199090)) 119905 isin [0 119879) (55)
Therefore along the trajectory120593(119905 1199090) (4) and the second
equation of (3) become
1198721015840(119905) = minus
120590
2
1198722(119905) + 120582119872 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
1205741015840(119905) = minus120574119872 ae 119905 isin [0 119879)
(56)
where the notation denotes the derivative with respect to 119905
and 119891 represents the function
119891 = minus119860119906 minus 120582120597minus1
119909119906 + 119892 (119905)
= minus119860119906 minus 120582120597minus1
119909119906 minus int
119878
(
120590
2
1199062
119909+
1
2
1205882minus 119860119906)119889119909
(57)
We first compute the upper and lower bounds for 119891 forlater use in getting the blow-up result as follows
119891 = minus119860119906 minus 120582int
119909
0
119906 119889119909 minus
120590
2
int
119878
1199062
119909119889119909 minus
1
2
int
119878
1205882119889119909 + 119860int
119878
119906 119889119909
le minus119860119906 minus 120582int
119909
0
119906 119889119909 + 119860int
119878
119906 119889119909 le
119860
2
(1 + 1199062)
+
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
(58)
Since 1199062 le (12) int119878(1199062+ 1199062
119909)119889119909 (17) we obtain the upper
bound for 119891
119891 le
119860
2
(1 +
1
2
int
119878
(1199062+ 1199062
119909) 119889119909) +
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le 119860 minus
120582
2
+
119860
4
int
119878
(1205882+ 1199062
119909) 119889119909 +
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860
4
1198640+
3119860 minus 2120582
4
[1198901198622119905(int
119878
1199062
0(119909) 119889119909 + 1)]
le 119860 minus
120582
2
+
119860
4
1198640+
3119860 minus 2120582
4
times [1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)] =
1
2
1198702
1(119879)
(59)
Now we turn to the lower bound of 119891 Using previous argu-ments we get
minus119891 = 119860119906 + 120582int
119909
0
119906 119889119909 +
120590
2
int
119878
1199062
119909119889119909 +
1
2
int
119878
1205882119889119909 minus 119860int
119878
119906 119889119909
le 119860
1 + 1199062
2
+
max (|120590| 1)2
int
119878
(1205882+ 1199062
119909) 119889119909
+
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le 119860 minus
120582
2
+
119860 + 2max (|120590| 1)4
int
119878
(1205882+ 1199062
119909) 119889119909
+
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860 + 2max (|120590| 1)4
1198640
+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(60)
When 120590 lt 0 we have a finer estimate
minus119891 le 119860 minus
120582
2
+
119860 + 2
4
int
119878
(1205882+ 1199062
119909) 119889119909 +
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860 + 2
4
1198640+
3119860 minus 2120582
4
times [1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)] =
1
2
1198702
2(119879)
(61)
Combining (59) and (60) we obtain
10038161003816100381610038161198911003816100381610038161003816le 119860 minus
120582
2
+
119860 + 2max (|120590| 1)4
1198640
+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)] =
1
2
1198702
3(119879)
(62)
Since 119904 ge 3 we have 119906 isin 11986210(119878) Therefore
sup119909isin119878
119906119909(119905 119909) ge 0 inf
119909isin119878
119906119909(119905 119909) le 0 119905 isin [0 119879) (63)
Journal of Applied Mathematics 7
Hence119872(119905) gt 0 for 119905 isin [0 119879) From the second equationof (55) we obtain
120574 (119905) = 120574 (0) 119890minusint119905
0
119872(120591)119889120591 (64)
Hence1003816100381610038161003816120588 (119905 120593 (119905 119909
0))1003816100381610038161003816=1003816100381610038161003816120574 (119905)
1003816100381610038161003816le1003816100381610038161003816120574 (0)
1003816100381610038161003816le10038171003817100381710038171205880
1003817100381710038171003817119871infin(119878) (65)
For any given 119909 isin 119878 define
1199011(119905) = 119872 (119905) minus
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
minusradic
10038171003817100381710038171205880
1003817100381710038171003817
2
119871infin(119878)+ 1198702
1(119879)
120590
+
1205822
1205902minus
120582
120590
(120590 gt 0)
(66)
Observing that 1199011(119905) is a 1198621-differentiable function on
[0 119879) and satisfies
1199011(0) = 119872 (0) minus
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
minusradic
10038171003817100381710038171205880
1003817100381710038171003817
2
119871infin(119878)+ 1198702
1(119879)
120590
+
1205822
1205902minus
120582
120590
le 119872(0) minus10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
(67)
We now claim that 1199011(119905) le 0 119905 isin [0 119879)
Assume the contrary that there is 1199050isin [0 119879) such that
1199011(1199050) gt 0
Let 1199051= max119905 lt 119905
0 1199011(119905) = 0 Then 119901
1(1199051) = 0 and
1199011015840
1(1199051) ge 0 or equivalently
119872(1199051) =
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)+radic
10038171003817100381710038171205880
1003817100381710038171003817
2
119871infin(119878)+ 1198702
1(119879)
120590
+
1205822
1205902+
120582
120590
(68)
and1198721015840(1199051) ge 0 ae 119905 isin [0 119879) On the other hand we have
1198721015840(1199051) = minus
120590
2
1198722(1199051) + 120582119872(119905
1) +
1
2
1205742(1199051)
+ 119891 (1199051 120593 (1199051 1199090)) ae 119905 isin [0 119879)
le minus
120590
2
(119872(1199051) minus
120582
120590
)
2
+
1
2
10038171003817100381710038171205880
1003817100381710038171003817
2
119871infin(119878)+
1205822
2120590
+
1198702
1(119879)
2
lt 0
(69)
which is a contradictionTherefore1199011(119905) le 0 for all 119905 isin [0 119879)
Since 119909 is arbitrarily chosen we obtain (45)To derive (46) in the case of 120590 lt 0 we consider (119905) and
120574(119905) as in Lemma 13
(119905) = 119906119909(119905 120577 (119905)) = inf
119909isin119878
[119906119909(119905 119909)] 119905 isin [0 119879) (70)
Hence
119906119909119909(119905 120577 (119905)) = 0 ae 119905 isin [0 119879) (71)
Using previous arguments we take the characteristic 120593(119905119909) defined in (13) and choose 119909
1(119905) isin 119877 such that
120593 (119905 1199091(119905)) = 120577 (119905) (72)
Let
120574 (119905) = 120588 (119905 120593 (119905 119909)) 119905 isin [0 119879) (73)
Hence along the trajectory 1015840(119905) = 120582(119905) +(12)1205742(119905) +
119891(119905 120593(119905 1199090)) ge 120582(0)+(12)120574
2(0) +(12)119870
2
2(119879) (4) and the
second equation of (3) become
1015840(119905) = minus
120590
2
2(119905) + 120582 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
1205741015840(119905) = minus120574 ae 119905 isin [0 119879)
(74)
Define
1199012(119905) = (119905) +
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
+radic1205822
1205902minus
1198702
2(119879)
120590
minus
120582
120590
(120590 lt 0)
(75)
For any given 119909 isin 119878 Note that 1199012(119905) is also 119862
1-dif-ferentiable function on [0 119879) and satisfies
1199012(0) = (0) +
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
+radic1205822
1205902minus
1198702
2(119879)
120590
minus
120582
120590
ge (0) +10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
ge 0
(76)
We now claim that 1199012(119905) ge 0 for any 119905 isin [0 119879)
Suppose not then there is isin [0 119879) such that 1199012(1199050) lt 0
Define
1199052= max 119905 lt 119905
0 1199012(119905) = 0 (77)
Then 1199012(1199052) = 0 and 1199011015840
2(1199052) lt 0 or equivalently
(1199052) = minus
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)minusradic1205822
1205902minus
1198702
2(119879)
120590
+
120582
120590
(78)
and 1015840(1199052) le 0 ae 119905 isin [0 119879) However we have
1015840(1199052) = minus
120590
2
2(1199052) + 120582 (119905
2) +
1
2
1205742(1199052)
+ 119891 (1199052 120593 (1199052 1199090)) ae 119905 isin [0 119879)
ge minus
120590
2
(minus10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)minusradic1205822
1205902minus
1198702
2(119879)
120590
)
2
+
1205822
2120590
minus
1
2
1198702
2(119879) gt 0
(79)
8 Journal of Applied Mathematics
Therefore 1199012(119905) ge 0 for any 119905 isin [0 119879) Since 119909 is chosen
arbitrarily we obtain (46)Let 120590 = 0 Using previous arguments (56) becomes
1198721015840(119905) = minus
120590
2
1198722(119905) + 120582119872 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
1205741015840(119905)
= minus120574119872 ae 119905 isin [0T) (80)
where the notation denotes the derivative with respect to 119905
and 119891 represents the function
119891 = minus119860119906 minus 120582120597minus1
119909119906 + 119892 (119905)
= minus119860119906 minus 120582120597minus1
119909119906 minus int
119878
(
1
2
1205882minus 119860119906)119889119909
(81)
We first compute the upper and lower bounds for 119891 forlater use in getting the blow-up result
119891 = minus119860119906 minus 120582int
119909
0
119906 119889119909 minus
1
2
int
119878
1205882119889119909 + 119860int
119878
119906 119889119909
le minus119860119906 minus 120582int
119909
0
119906 119889119909 + 119860int
119878
119906 119889119909 le
119860
2
(1 + 1199062)
+
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le
119860
2
(1 +
1
2
int
119878
(1199062+ 1199062
119909) 119889119909) +
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le 119860 minus
120582
2
+
119860
4
int
119878
(1205882+ 1199062
119909) 119889119909 +
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860
4
1198640+
3119860 minus 2120582
4
[1198901198622119905(int
119878
1199062
0(119909) 119889119909 + 1)]
le 119860 minus
120582
2
+
119860
4
1198640+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(82)
Now we turn to the lower bound of 119891
minus119891 = 119860119906 + 120582int
119909
0
119906 119889119909 +
1
2
int
119878
1205882119889119909 minus 119860int
119878
119906 119889119909
le 119860
1 + 1199062
2
+
1
2
int
119878
1205882119889119909 +
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le 119860 minus
120582
2
+
119860 + 2
4
int
119878
(1205882+ 1199062
119909) 119889119909 +
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860 + 2
4
1198640+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(83)
Combining (82) and (83) we obtain
10038161003816100381610038161198911003816100381610038161003816le 119860 minus
120582
2
+
119860 + 2
4
1198640+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(84)
We know 119872(119905) gt 0 for 119905 isin [0 119879) From the secondequation of (81) we obtain that
120574 (119905) = 120574 (0) 119890minusint119905
0
119872(120591)119889120591 (85)
Hence1003816100381610038161003816120588 (119905 120593 (119905 119909
0))1003816100381610038161003816=1003816100381610038161003816120574 (119905)
1003816100381610038161003816le1003816100381610038161003816120574 (0)
1003816100381610038161003816 (86)
Therefore we have
1198721015840(119905) = 120582119872 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
le 120582119872 (0) +
1
2
1205742(0) +
1
2
1198702
1(119879)
(87)
1198721015840(119905) minus 120582119872 (119905) le
1
2
(sup119909isin119878
1205882
0(119909) + 119870
2
1(119879)) ae 119905 isin [0 119879)
(88)
Integrating (88) on [0 119905] we prove (47) as follows
119872(119905) le sup119909isin119878
1199060119909
(119909) +
1
2
(sup119909isin119878
1205882
0(119909) + 119870
2
1(119879))
119890120582119905minus 1
120582
(89)
To obtain a lower bound for inf119909isin119878
119906119909(119905 119909) we use the
same argumentSince 120590 = 0 (80) becomes
1015840(119905) = 120582 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
1))
1205741015840(119905) = minus120574 ae 119905 isin [0 119879)
(90)
Because of (119905) lt 0 we get from the second equation of(90) that
120574 (119905) = 120574 (0) 119890minusint119905
0
119872(120591)119889120591 (91)
This means that1003816100381610038161003816120588 (119905 120593 (119905 119909
1))1003816100381610038161003816=1003816100381610038161003816120574 (119905)
1003816100381610038161003816ge1003816100381610038161003816120574 (0)
1003816100381610038161003816 (92)
Then
1015840(119905) = 120582 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
ge 120582 (0) +
1
2
1205742(0) +
1
2
1198702
2(119879)
(93)
1015840(119905) minus 120582 (119905) ge
1
2
(inf119909isin119878
1205882
0(119909) minus 119870
2
2(119879)) ae 119905 isin [0 119879)
(94)
Integrating (94) on [0 119905] we prove (48) This completesthe proof of Lemma 14
Lemma 15 Suppose that 120590 isin 119877 0 Suppose 1198830= (1199060
1205880
) isin
119867119904times119867119904minus1 with 119904 ge 2 and let 119879 be the maximal existence time
Journal of Applied Mathematics 9
of the solution 119883 = (119906
120588 ) to (3) with initial data 1198830 Then we
have
120588 (119905 120593 (119905 119909)) 120593119909(119905 119909) = 120588
0(119909) (119905 119909) isin [0 119879] times 119878 (95)
Moreover if there exists119872 gt 0 such that
inf(119905119909)isin[0119879]times119878
120590119906119909(119905 119909) ge minus119872 (119905 119909) isin [0 119879] times 119878 (96)
Then1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin(119878)
=1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin(119878)
le 119890119872119879120590 1003817
1003817100381710038171205880
1003817100381710038171003817119871infin(119878)
(120590 gt 0)
(97)
1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin(119878)
=1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin(119878)
le 119890119873119879120590 1003817
1003817100381710038171205880
1003817100381710038171003817119871infin(119878)
(120590 lt 0)
(98)
where119873 = 1199060119909119871infin(119878)+radic12058221205902minus 1198702
2(119879)120590 minus 120582120590 and119870
2(119879)
is given in (50)
Proof Differentiating the left hand side of (95) with respectto 119905 in view of the relations (12) and (3) we obtain
119889
119889119905
120588 (119905 120593 (119905 119909)) 120593119909(119905 119909)
= [120588119905(119905 120593) + 120588
119909(119905 120593) 120593
119905(119905 119909)]
times 120593119909(119905 119909) + 120588 (119905 120593) 120593
119909119905(119905 119909)
= [120588119905(119905 120593) + 120588
119909(119905 120593) 120593
119905(119905 119909)] 120593
119909(119905 119909)
+ 120588 (119905 120593) 119906119909(119905 120593) 120593
119909(119905 119909)
= [120588119905(119905 120593) + 120588
119909(119905 120593) 120593
119905(119905 119909) + 120588 (119905 120593) 119906
119909(119905 120593)]
times 120593119909(119905 119909) = 0
(99)
This completes the proof of (95) In view of the assump-tion (96) and 120590 gt 0 we obtain 119906(119905 119909) ge minus(119872120590) (119905 119909) isin
[0 119879] times 119878By Lemma 3 and (95) we have
1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin =
1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin =
1003817100381710038171003817100381710038171003817
119890minusint119905
0
119906119909(120591sdot)119889120591
1205880(sdot)
1003817100381710038171003817100381710038171003817119871infin
le 1198901198721198791205901003817
1003817100381710038171205880(sdot)1003817100381710038171003817119871infin
(100)
To obtain (98) we use a similar argument as before Using(13) and the lower bound for 119906
119909(119905 119909) in (46) it follows that
1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin =
1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin =
1003817100381710038171003817100381710038171003817
119890minusint119905
0
119906119909(120591sdot)119889120591
1205880(sdot)
1003817100381710038171003817100381710038171003817119871infin
le 1198901198731198791205901003817
1003817100381710038171205880(sdot)1003817100381710038171003817119871infin
(101)
which proves (98) This completes the proof of Lemma 15
Proof of Theorem 12 Suppose that119879 lt infin and that (42) is notvalid Then there is some positive number119872 gt 0 such that
120590119906119909(119905 119909) ge minus119872 (119905 119909) isin [0 119879] times 119878 (102)
It now follows from Lemma 14 that |119906119909(119905 119909)| le 119862 where
119862 = 119862(119860119872 120590 1198640 120582 119906
0 119879) Therefore Theorem 11 implies
that the maximal existence time 119879 = infin which contradictswith the assumption that 119879 lt infin
Conversely the Sobolev embedding theorem 119867119904(119878) rarr
119871infin(119878) with 119904 gt 12 implies that if (70) holds the corre-
sponding solution blows up in finite time which completesthe proof of Theorem 12
Acknowledgments
The authors would like to thank the referee for commentsand suggestions This work is supported by the NationalNature Science Foundation of China (no 11171135) theNature Science Foundation of the Jiangsu Higher EducationInstitutions of China (no 09KJB110003) and the high-leveltalented person special subsidizes of Jiangsu University (no05JDG047)
References
[1] J K Hunter and R Saxton ldquoDynamics of director fieldsrdquo SIAMJournal on Applied Mathematics vol 51 no 6 pp 1498ndash15211991
[2] Z Yin ldquoOn the structure of solutions to the periodic Hunter-Saxton equationrdquo SIAM Journal on Mathematical Analysis vol36 no 1 pp 272ndash283 2004
[3] H-H Dai and M Pavlov ldquoTransformations for the Camassa-Holm equation its high-frequency limit and the Sinh-Gordonequationrdquo Journal of the Physical Society of Japan vol 67 no 11pp 3655ndash3657 1998
[4] J K Hunter and Y X Zheng ldquoOn a completely integrablenonlinear hyperbolic variational equationrdquo Physica D vol 79no 2ndash4 pp 361ndash386 1994
[5] R Camassa and D D Holm ldquoAn integrable shallow waterequation with peaked solitonsrdquo Physical Review Letters vol 71no 11 pp 1661ndash1664 1993
[6] R S Johnson ldquoCamassa-Holm Korteweg-de Vries and relatedmodels for water wavesrdquo Journal of Fluid Mechanics vol 455pp 63ndash82 2002
[7] A Constantin and B Kolev ldquoOn the geometric approach to themotion of inertialmechanical systemsrdquo Journal of Physics A vol35 no 32 pp R51ndashR79 2002
[8] P J Olver and P Rosenau ldquoTri-Hamiltonian duality betweensolitons and solitary-wave solutions having compact supportrdquoPhysical Review E vol 53 no 2 pp 1900ndash1906 1996
[9] R Beals D H Sattinger and J Szmigielski ldquoInverse scatteringsolutions of the Hunter-Saxton equationrdquo Applicable Analysisvol 78 no 3-4 pp 255ndash269 2001
[10] A Bressan and A Constantin ldquoGlobal solutions of the Hunter-Saxton equationrdquo SIAM Journal on Mathematical Analysis vol94 pp 68ndash92 2010
[11] A Bressan H Holden and X Raynaud ldquoLipschitz metric forthe Hunter-Saxton equationrdquo Journal de Mathematiques Pureset Appliquees vol 94 no 1 pp 68ndash92 2010
10 Journal of Applied Mathematics
[12] J K Hunter and Y X Zheng ldquoOn a nonlinear hyperbolicvariational equation I Global existence of weak solutionsrdquoArchive for Rational Mechanics and Analysis vol 129 no 4 pp305ndash353 1995
[13] J Lenells ldquoThe Hunter-Saxton equation describes the geodesicflow on a sphererdquo Journal of Geometry and Physics vol 57 no10 pp 2049ndash2064 2007
[14] J B Li and Y S Li ldquoBifurcations of travelling wave solutions fora two-component Camassa-Holm equationrdquoActaMathematicaSinica (English Series) vol 24 no 8 pp 1319ndash1330 2008
[15] W Rui and Y Long ldquoIntegral bifurcation method together witha translation-dilation transformation for solving an integrable2-component Camassa-Holm shallow water systemrdquo Journal ofApplied Mathematics vol 2012 Article ID 736765 21 pages2012
[16] H Wu and M Wunsch ldquoGlobal existence for the generalizedtwo-component Hunter-Saxton systemrdquo Journal of Mathemati-cal Fluid Mechanics vol 14 no 3 pp 455ndash469 2012
[17] BMoon andY Liu ldquoWave breaking and global existence for thegeneralized periodic two-component Hunter-Saxton systemrdquoJournal of Differential Equations vol 253 no 1 pp 319ndash3552012
[18] S Wu and Z Yin ldquoBlow-up blow-up rate and decay of thesolution of the weakly dissipative Camassa-Holm equationrdquoJournal ofMathematical Physics vol 47 no 1 Article ID 0135042006
[19] S Wu J Escher and Z Yin ldquoGlobal existence and blow-up phenomena for a weakly dissipative Degasperis-Procesiequationrdquo Discrete and Continuous Dynamical Systems B vol12 no 3 pp 633ndash645 2009
[20] T Kato ldquoQuasi-linear equations of evolution with applicationsto partial differential equationsrdquo in Spectral Theory and Differ-ential Equations vol 448 of Lecture Notes in Mathematics pp25ndash70 Springer Berlin Germany 1975
[21] G Gui and Y Liu ldquoOn the global existence and wave-breakingcriteria for the two-component Camassa-Holm systemrdquo Journalof Functional Analysis vol 258 no 12 pp 4251ndash4278 2010
[22] A Constantin and J Escher ldquoWave breaking for nonlinearnonlocal shallow water equationsrdquo Acta Mathematica vol 181no 2 pp 229ndash243 1998
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Applied Mathematics
Multiplying (6) by 119906 and integrating with respect to 119909using the periodicity of 119906 and (24) we obtain
1
2
119889
119889119905
int
119878
1199062(119905 119909) 119889119909
= int
119878
119906119906119905119889119909
= minus120590int
119878
1199061199091199062119889119909
+ int
119878
119906 [120597minus1
119909(
120590
2
1199062
119909+
1
2
1205882minus 119860119906 + 120582119906
119909
minus 120582120597minus1
119909119906 + 119892) +ℎ (119905) ] 119889119909
= int
119878
119906 [120597minus1
119909(
120590
2
1199062
119909+
1
2
1205882minus 119860119906 + 120582119906
119909
minus120582120597minus1
119909119906 + 119892) + ℎ (119905) ] 119889119909
le (1198621+ (2 |119860| minus 120582) int
119878
|119906| 119889119909 minus 120582)int
119878
|119906| 119889119909
le 1198621int
119878
|119906| 119889119909 + (2 |119860| minus 120582) (int
119878
|119906| 119889119909)
2
minus 120582int
119878
1199062119889119909
le (
1198621
2
+ (2 |119860| minus 2120582))int
119878
1199062119889119909 +
1198621minus 120582
2
=
1198622
2
int
119878
1199062119889119909 +
1198621minus 120582
2
(26)
where 1198622= 1198621+ 4119860 minus 2120582 note that 119862
2gt 1198621
By Gronwallrsquos inequality we get
int
119878
1199062(119905 119909) 119889119909 le 119890
1198622119905(int
119878
1199062
0(119909) 119889119909 +
1198621minus 120582
1198622
) minus
1198621minus 120582
1198622
le 1198901198622119905(int
119878
1199062
0(119909) 119889119909 + 1)
(27)
This completes the proof of Lemma 6
Lemma 7 Assume that 1199060isin 119867119904(119878) 119904 ge 2 119906
0= 0 and that
the corresponding solution 119906(119905 119909) of (3) has a zero point forany time 119905 ge 0 Then for all 119905 isin [0 119879) we have
int
119878
1199062(119905 119909) 119889119909 le int
119878
1199062
119909(119905 119909) 119889119909 le 119864
0 (28)
Proof By assumption there is 1199090isin [0 1] such that 119906(119905 119909
0) =
0 for each 119905 isin [0 119879)Then for 119909 isin 119878 by holder equality we have
1199062(119905 119909) = (int
119909
1199090
119906119909119889119909)
2
le (119909 minus 1199090) int
119909
1199090
1199062
119909119889119909
119909 isin [1199090 1199090+
1
2
]
(29)
This implies sup119909isin119878
1199062(119905 119909) le (12) int
1198781199062
119909119889119909
int
119878
1199062(119905 119909) 119889119909 le sup
119909isin119878
1199062(119905 119909) le
1
2
int
119878
1199062
119909119889119909
le int
119878
1199062
119909+ 1205882119889119909 le 119864
0
(30)
3 Wave-Breaking Criteria
In this section by using transport equation theory we obtainthe wave-breaking criteria for solutions to (3) We first recallthe following propositions
Proposition 8 (1D Moser-type estimates) The following esti-mates hold
(a) For 119904 ge 010038171003817100381710038171198911198921003817100381710038171003817119867119904(R) le 119862 (
10038171003817100381710038171198911003817100381710038171003817119871infin(R)10038171003817100381710038171198921003817100381710038171003817119867119904(R) +
10038171003817100381710038171198911003817100381710038171003817119867119904(R)10038171003817100381710038171198921003817100381710038171003817119871infin(R)) (31)
(b) For 119904 gt 010038171003817100381710038171198911205971199091198921003817100381710038171003817119867119904(R) le 119862 (
10038171003817100381710038171198911003817100381710038171003817119871infin(R)10038171003817100381710038171205971199091198921003817100381710038171003817119867119904(R) +
10038171003817100381710038171198911003817100381710038171003817119867119904+1(R)10038171003817100381710038171198921003817100381710038171003817119871infin(R))
(32)
(c) For 1199041le 12 119904
2gt 12 119904
1+ 1199042gt 0
1003817100381710038171003817f11989210038171003817100381710038171198671199041 (R) le 119862
100381710038171003817100381711989110038171003817100381710038171198671199041 (R)
100381710038171003817100381711989210038171003817100381710038171198671199042 (R) (33)
where 1198621015840119904 are constants that are independent of 119891 and 119892
Proposition 9 (see [21]) Suppose that 119904 gt minus1198892 Let Vbe a vector field such that nablaV belongs to 119871
1([0 119879]119867
119904minus1) if
119904 gt 1 + 1198892 or to 1198711([0 119879]1198671198892 cap 119871infin) otherwise Suppose
also that 1198910
isin 119867119904 119865 isin 119871
1([0 119879]119867
119904) and that 119891 isin
119871infin([0 119879]119867
119904) cap 119862([0 119879] 119878
1015840) solves the 119889-dimensional linear
transport equations
(119879)
120597119905119891 + V sdot nabla119891 = 119865
1198911003816100381610038161003816119905=0
= 1198910
(34)
Then 119891 isin 119862([0 119879]119867119904) More precisely there exists a
constant119862 depending only on 119904119901 and119889 such that the followingstatements hold
(1) If 119904 = 1 + 1198892
10038171003817100381710038171198911003817100381710038171003817119867119904 le
10038171003817100381710038171198910
1003817100381710038171003817119867119904 + int
119905
0
119865 (120591)119867119904119889120591 + 119862int
119905
0
1198811015840(120591)
1003817100381710038171003817119891 (120591)
1003817100381710038171003817119867119904119889120591
(35)
or
10038171003817100381710038171198911003817100381710038171003817119867119904 le 119890119862119881(119905)
(10038171003817100381710038171198910
1003817100381710038171003817119867119904 + int
119905
0
119890minus119862119881(120591)
119865 (120591)119867119904119889120591) (36)
with 119881(119905) = int
119905
0nablaV(120591)
1198671198892cap119871infin119889120591 if 119904 lt 1 + 1198892 and 119881(119905) =
int
119905
0nablaV(120591)
119867119904minus1119889120591 else
Journal of Applied Mathematics 5
(2) If 119891 = V then for all 119904 gt 0 estimates (35) and (36) holdwith
119881 (119905) = int
119905
0
1003817100381710038171003817120597119909119906 (120591)
1003817100381710038171003817119871infin119889120591 (37)
Proposition 10 (see [21]) Let 0 lt 119904 lt 1 Suppose that 1198910isin
119867119904 119892 isin 119871
1([0 119879]119867
119904)V 120597119909V isin 119871
1([0 119879] 119871
infin) and that 119891 isin
119871infin([0 119879]119867
119904) cap 119862([0 119879] 119878
1015840) solves the 1-dimensional linear
transport equation
(119879)
120597119905119891 + V sdot nabla119891 = 119892
1198911003816100381610038161003816119905=0
= 1198910
(38)
Then 119891 isin 119862([0 119879]H119904) More precisely there exists a con-stant 119862 depending only on 119904 such that the following statementshold
10038171003817100381710038171198911003817100381710038171003817119867119904 le
10038171003817100381710038171198910
1003817100381710038171003817119867119904 + 119862int
119905
0
1003817100381710038171003817119892 (120591)
1003817100381710038171003817119867119904119889120591
+ 119862int
119905
0
1198811015840(120591)
1003817100381710038171003817119891 (120591)
1003817100381710038171003817119867119904119889120591
(39)
or
10038171003817100381710038171198911003817100381710038171003817119867119904 le 119890119862119881(119905)
(10038171003817100381710038171198910
1003817100381710038171003817119867119904 + 119862int
119905
0
1003817100381710038171003817119892 (120591)
1003817100381710038171003817119867119904119889120591) (40)
with 119881(119905) = int
119905
0(V(120591)
119871infin + 120597
119909V(120591)119871infin)119889120591
The above proposition was proved in [8] using Little-wood-Paley analysis for the transport equation and Moser-type estimates Using this result and performing the sameargument as in [17] we can obtain the following blow-upcriterion
Theorem 11 Let1198830= (1199060
1205880
) isin 119867119904times119867119904minus1 with 119904 ge 2 and119883 =
(119906
120588 ) be the corresponding solution to (3) Assume that 119879 gt 0 isthe maximal time of existence Then
119879 lt infin 997904rArr int
119879
0
1003817100381710038171003817120597119909119906 (120591)
1003817100381710038171003817119871infin119889120591 = infin (41)
Our next result describes the necessary and sufficientcondition for the blow-up of solutions to (3)
Theorem 12 Suppose that 120590 isin 119877 0 Let 1198830= (1199060
1205880
) isin 119867119904times
119867119904minus1 with 119904 ge 2 and let119879 be themaximal existence time of the
solution119883 = (119906
120588 ) to (3)with initial data1198830 Then the solutionblows up in finite time if and only if
lim inf119905rarr119879
minus
inf119909isin119878
120590119906119909(119905 119909) = minusinfin (42)
The approach one takes here is the method of charac-teristics Applying the following lemma we may carry outthe estimates along the characteristics 120593(119905 119909) which capturessup119909isin119878
119906119909(119905 119909) and inf
119909isin119878119906119909(119905 119909)
Lemma 13 (see [22]) Let119879 gt 0 and let V isin 1198621([0 119879]1198672(119877))Then for every 119905 isin [0 119879) there exists at least one point 120585(119905) isin 119877with
119898(119905) = inf119909isin119878
V119909(119905 119909) = V
119909(119905 120585 (119905)) (43)
and the function 119898(119905) is almost everywhere differentiable on(0 119879) with
119889119898 (119905)
119889119905
= V119905119909(119905 120585 (119905)) 119886119890 119900119899 (0 119879) (44)
Lemma 14 Let 1198830= (1199060
1205880
) isin 119867119904times 119867119904minus1 with 119904 ge 2 and let 119879
be the maximal existence time of the solution 119883 = (119906
120588 ) to (3)with initial data119883
0 Then one has the following
(1) 120590 = 0
sup119909isin119878
119906119909(119905 119909) le
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
+radic
10038171003817100381710038171205880
1003817100381710038171003817
2
119871infin(119878)+ 1198702
1(119879)
120590
+
1205822
1205902+
120582
120590
(120590 gt 0)
(45)
inf119909isin119878
119906119909(119905 119909) ge minus
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
minusradic1205822
1205902minus
1198702
2(119879)
120590
+
120582
120590
(120590 lt 0)
(46)
(2) 120590 = 0
sup119909isin119878
119906119909(119905 119909) le sup
119909isin119878
1199060119909
(119909)
+
1
2
(sup119909isin119878
1205882
0(119909) + 119870
2
1(119879))
119890120582119905minus 1
120582
(47)
inf119909isin119878
119906119909(119905 119909) ge inf
119909isin119878
1199060119909
(119909)
+
1
2
(inf119909isin119878
1205882
0(119909) minus 119870
2
2(119879))
119890120582119905minus 1
120582
(48)
The constants above are defined as follows
1198701(119879) = radic2119860 minus 120582 +
119860
2
1198640+
3119860 minus 2120582
2
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(49)
1198702(119879)
= radic2119860 minus 120582 +
119860 + 2
2
1198640+
3119860 minus 2120582
2
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(50)
Proof of Lemma 14 ByTheorem 2 and a simple density argu-ment we show that the desired results are valid when 119904 ge 3so we take 119904 = 3 in the proof
6 Journal of Applied Mathematics
Let 120590 gt 0 Using Lemma 13 and the fact that
sup119909isin119878
[V119909(119905 119909)] = minusinf
119909isin119878
[minusV119909(119905 119909)] (51)
We can consider119872(119905) and 120574(119905) as follows
119872(119905) = 119906119909(119905 120585 (119905)) = sup
119909isin119878
[119906119909(119905 119909)] 119905 isin [0 119879) (52)
Hence
119906119909119909(119905 120585 (119905)) = 0 ae on 119905 isin [0 119879) (53)
Take the trajectory 120593(119905 119909) defined in (12) Then we knowthat 120593(119905 sdot) 119877 rarr 119877 is a diffeomorphism for every 119905 isin [0 119879)Therefore there exists 119909
0(119905) isin 119877 such that
120593 (119905 1199090(119905)) = 120585 (119905) 119905 isin [0 119879) (54)
Now let
120574 (119905) = 120588 (119905 120593 (119905 1199090)) 119905 isin [0 119879) (55)
Therefore along the trajectory120593(119905 1199090) (4) and the second
equation of (3) become
1198721015840(119905) = minus
120590
2
1198722(119905) + 120582119872 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
1205741015840(119905) = minus120574119872 ae 119905 isin [0 119879)
(56)
where the notation denotes the derivative with respect to 119905
and 119891 represents the function
119891 = minus119860119906 minus 120582120597minus1
119909119906 + 119892 (119905)
= minus119860119906 minus 120582120597minus1
119909119906 minus int
119878
(
120590
2
1199062
119909+
1
2
1205882minus 119860119906)119889119909
(57)
We first compute the upper and lower bounds for 119891 forlater use in getting the blow-up result as follows
119891 = minus119860119906 minus 120582int
119909
0
119906 119889119909 minus
120590
2
int
119878
1199062
119909119889119909 minus
1
2
int
119878
1205882119889119909 + 119860int
119878
119906 119889119909
le minus119860119906 minus 120582int
119909
0
119906 119889119909 + 119860int
119878
119906 119889119909 le
119860
2
(1 + 1199062)
+
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
(58)
Since 1199062 le (12) int119878(1199062+ 1199062
119909)119889119909 (17) we obtain the upper
bound for 119891
119891 le
119860
2
(1 +
1
2
int
119878
(1199062+ 1199062
119909) 119889119909) +
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le 119860 minus
120582
2
+
119860
4
int
119878
(1205882+ 1199062
119909) 119889119909 +
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860
4
1198640+
3119860 minus 2120582
4
[1198901198622119905(int
119878
1199062
0(119909) 119889119909 + 1)]
le 119860 minus
120582
2
+
119860
4
1198640+
3119860 minus 2120582
4
times [1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)] =
1
2
1198702
1(119879)
(59)
Now we turn to the lower bound of 119891 Using previous argu-ments we get
minus119891 = 119860119906 + 120582int
119909
0
119906 119889119909 +
120590
2
int
119878
1199062
119909119889119909 +
1
2
int
119878
1205882119889119909 minus 119860int
119878
119906 119889119909
le 119860
1 + 1199062
2
+
max (|120590| 1)2
int
119878
(1205882+ 1199062
119909) 119889119909
+
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le 119860 minus
120582
2
+
119860 + 2max (|120590| 1)4
int
119878
(1205882+ 1199062
119909) 119889119909
+
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860 + 2max (|120590| 1)4
1198640
+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(60)
When 120590 lt 0 we have a finer estimate
minus119891 le 119860 minus
120582
2
+
119860 + 2
4
int
119878
(1205882+ 1199062
119909) 119889119909 +
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860 + 2
4
1198640+
3119860 minus 2120582
4
times [1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)] =
1
2
1198702
2(119879)
(61)
Combining (59) and (60) we obtain
10038161003816100381610038161198911003816100381610038161003816le 119860 minus
120582
2
+
119860 + 2max (|120590| 1)4
1198640
+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)] =
1
2
1198702
3(119879)
(62)
Since 119904 ge 3 we have 119906 isin 11986210(119878) Therefore
sup119909isin119878
119906119909(119905 119909) ge 0 inf
119909isin119878
119906119909(119905 119909) le 0 119905 isin [0 119879) (63)
Journal of Applied Mathematics 7
Hence119872(119905) gt 0 for 119905 isin [0 119879) From the second equationof (55) we obtain
120574 (119905) = 120574 (0) 119890minusint119905
0
119872(120591)119889120591 (64)
Hence1003816100381610038161003816120588 (119905 120593 (119905 119909
0))1003816100381610038161003816=1003816100381610038161003816120574 (119905)
1003816100381610038161003816le1003816100381610038161003816120574 (0)
1003816100381610038161003816le10038171003817100381710038171205880
1003817100381710038171003817119871infin(119878) (65)
For any given 119909 isin 119878 define
1199011(119905) = 119872 (119905) minus
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
minusradic
10038171003817100381710038171205880
1003817100381710038171003817
2
119871infin(119878)+ 1198702
1(119879)
120590
+
1205822
1205902minus
120582
120590
(120590 gt 0)
(66)
Observing that 1199011(119905) is a 1198621-differentiable function on
[0 119879) and satisfies
1199011(0) = 119872 (0) minus
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
minusradic
10038171003817100381710038171205880
1003817100381710038171003817
2
119871infin(119878)+ 1198702
1(119879)
120590
+
1205822
1205902minus
120582
120590
le 119872(0) minus10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
(67)
We now claim that 1199011(119905) le 0 119905 isin [0 119879)
Assume the contrary that there is 1199050isin [0 119879) such that
1199011(1199050) gt 0
Let 1199051= max119905 lt 119905
0 1199011(119905) = 0 Then 119901
1(1199051) = 0 and
1199011015840
1(1199051) ge 0 or equivalently
119872(1199051) =
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)+radic
10038171003817100381710038171205880
1003817100381710038171003817
2
119871infin(119878)+ 1198702
1(119879)
120590
+
1205822
1205902+
120582
120590
(68)
and1198721015840(1199051) ge 0 ae 119905 isin [0 119879) On the other hand we have
1198721015840(1199051) = minus
120590
2
1198722(1199051) + 120582119872(119905
1) +
1
2
1205742(1199051)
+ 119891 (1199051 120593 (1199051 1199090)) ae 119905 isin [0 119879)
le minus
120590
2
(119872(1199051) minus
120582
120590
)
2
+
1
2
10038171003817100381710038171205880
1003817100381710038171003817
2
119871infin(119878)+
1205822
2120590
+
1198702
1(119879)
2
lt 0
(69)
which is a contradictionTherefore1199011(119905) le 0 for all 119905 isin [0 119879)
Since 119909 is arbitrarily chosen we obtain (45)To derive (46) in the case of 120590 lt 0 we consider (119905) and
120574(119905) as in Lemma 13
(119905) = 119906119909(119905 120577 (119905)) = inf
119909isin119878
[119906119909(119905 119909)] 119905 isin [0 119879) (70)
Hence
119906119909119909(119905 120577 (119905)) = 0 ae 119905 isin [0 119879) (71)
Using previous arguments we take the characteristic 120593(119905119909) defined in (13) and choose 119909
1(119905) isin 119877 such that
120593 (119905 1199091(119905)) = 120577 (119905) (72)
Let
120574 (119905) = 120588 (119905 120593 (119905 119909)) 119905 isin [0 119879) (73)
Hence along the trajectory 1015840(119905) = 120582(119905) +(12)1205742(119905) +
119891(119905 120593(119905 1199090)) ge 120582(0)+(12)120574
2(0) +(12)119870
2
2(119879) (4) and the
second equation of (3) become
1015840(119905) = minus
120590
2
2(119905) + 120582 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
1205741015840(119905) = minus120574 ae 119905 isin [0 119879)
(74)
Define
1199012(119905) = (119905) +
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
+radic1205822
1205902minus
1198702
2(119879)
120590
minus
120582
120590
(120590 lt 0)
(75)
For any given 119909 isin 119878 Note that 1199012(119905) is also 119862
1-dif-ferentiable function on [0 119879) and satisfies
1199012(0) = (0) +
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
+radic1205822
1205902minus
1198702
2(119879)
120590
minus
120582
120590
ge (0) +10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
ge 0
(76)
We now claim that 1199012(119905) ge 0 for any 119905 isin [0 119879)
Suppose not then there is isin [0 119879) such that 1199012(1199050) lt 0
Define
1199052= max 119905 lt 119905
0 1199012(119905) = 0 (77)
Then 1199012(1199052) = 0 and 1199011015840
2(1199052) lt 0 or equivalently
(1199052) = minus
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)minusradic1205822
1205902minus
1198702
2(119879)
120590
+
120582
120590
(78)
and 1015840(1199052) le 0 ae 119905 isin [0 119879) However we have
1015840(1199052) = minus
120590
2
2(1199052) + 120582 (119905
2) +
1
2
1205742(1199052)
+ 119891 (1199052 120593 (1199052 1199090)) ae 119905 isin [0 119879)
ge minus
120590
2
(minus10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)minusradic1205822
1205902minus
1198702
2(119879)
120590
)
2
+
1205822
2120590
minus
1
2
1198702
2(119879) gt 0
(79)
8 Journal of Applied Mathematics
Therefore 1199012(119905) ge 0 for any 119905 isin [0 119879) Since 119909 is chosen
arbitrarily we obtain (46)Let 120590 = 0 Using previous arguments (56) becomes
1198721015840(119905) = minus
120590
2
1198722(119905) + 120582119872 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
1205741015840(119905)
= minus120574119872 ae 119905 isin [0T) (80)
where the notation denotes the derivative with respect to 119905
and 119891 represents the function
119891 = minus119860119906 minus 120582120597minus1
119909119906 + 119892 (119905)
= minus119860119906 minus 120582120597minus1
119909119906 minus int
119878
(
1
2
1205882minus 119860119906)119889119909
(81)
We first compute the upper and lower bounds for 119891 forlater use in getting the blow-up result
119891 = minus119860119906 minus 120582int
119909
0
119906 119889119909 minus
1
2
int
119878
1205882119889119909 + 119860int
119878
119906 119889119909
le minus119860119906 minus 120582int
119909
0
119906 119889119909 + 119860int
119878
119906 119889119909 le
119860
2
(1 + 1199062)
+
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le
119860
2
(1 +
1
2
int
119878
(1199062+ 1199062
119909) 119889119909) +
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le 119860 minus
120582
2
+
119860
4
int
119878
(1205882+ 1199062
119909) 119889119909 +
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860
4
1198640+
3119860 minus 2120582
4
[1198901198622119905(int
119878
1199062
0(119909) 119889119909 + 1)]
le 119860 minus
120582
2
+
119860
4
1198640+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(82)
Now we turn to the lower bound of 119891
minus119891 = 119860119906 + 120582int
119909
0
119906 119889119909 +
1
2
int
119878
1205882119889119909 minus 119860int
119878
119906 119889119909
le 119860
1 + 1199062
2
+
1
2
int
119878
1205882119889119909 +
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le 119860 minus
120582
2
+
119860 + 2
4
int
119878
(1205882+ 1199062
119909) 119889119909 +
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860 + 2
4
1198640+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(83)
Combining (82) and (83) we obtain
10038161003816100381610038161198911003816100381610038161003816le 119860 minus
120582
2
+
119860 + 2
4
1198640+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(84)
We know 119872(119905) gt 0 for 119905 isin [0 119879) From the secondequation of (81) we obtain that
120574 (119905) = 120574 (0) 119890minusint119905
0
119872(120591)119889120591 (85)
Hence1003816100381610038161003816120588 (119905 120593 (119905 119909
0))1003816100381610038161003816=1003816100381610038161003816120574 (119905)
1003816100381610038161003816le1003816100381610038161003816120574 (0)
1003816100381610038161003816 (86)
Therefore we have
1198721015840(119905) = 120582119872 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
le 120582119872 (0) +
1
2
1205742(0) +
1
2
1198702
1(119879)
(87)
1198721015840(119905) minus 120582119872 (119905) le
1
2
(sup119909isin119878
1205882
0(119909) + 119870
2
1(119879)) ae 119905 isin [0 119879)
(88)
Integrating (88) on [0 119905] we prove (47) as follows
119872(119905) le sup119909isin119878
1199060119909
(119909) +
1
2
(sup119909isin119878
1205882
0(119909) + 119870
2
1(119879))
119890120582119905minus 1
120582
(89)
To obtain a lower bound for inf119909isin119878
119906119909(119905 119909) we use the
same argumentSince 120590 = 0 (80) becomes
1015840(119905) = 120582 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
1))
1205741015840(119905) = minus120574 ae 119905 isin [0 119879)
(90)
Because of (119905) lt 0 we get from the second equation of(90) that
120574 (119905) = 120574 (0) 119890minusint119905
0
119872(120591)119889120591 (91)
This means that1003816100381610038161003816120588 (119905 120593 (119905 119909
1))1003816100381610038161003816=1003816100381610038161003816120574 (119905)
1003816100381610038161003816ge1003816100381610038161003816120574 (0)
1003816100381610038161003816 (92)
Then
1015840(119905) = 120582 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
ge 120582 (0) +
1
2
1205742(0) +
1
2
1198702
2(119879)
(93)
1015840(119905) minus 120582 (119905) ge
1
2
(inf119909isin119878
1205882
0(119909) minus 119870
2
2(119879)) ae 119905 isin [0 119879)
(94)
Integrating (94) on [0 119905] we prove (48) This completesthe proof of Lemma 14
Lemma 15 Suppose that 120590 isin 119877 0 Suppose 1198830= (1199060
1205880
) isin
119867119904times119867119904minus1 with 119904 ge 2 and let 119879 be the maximal existence time
Journal of Applied Mathematics 9
of the solution 119883 = (119906
120588 ) to (3) with initial data 1198830 Then we
have
120588 (119905 120593 (119905 119909)) 120593119909(119905 119909) = 120588
0(119909) (119905 119909) isin [0 119879] times 119878 (95)
Moreover if there exists119872 gt 0 such that
inf(119905119909)isin[0119879]times119878
120590119906119909(119905 119909) ge minus119872 (119905 119909) isin [0 119879] times 119878 (96)
Then1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin(119878)
=1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin(119878)
le 119890119872119879120590 1003817
1003817100381710038171205880
1003817100381710038171003817119871infin(119878)
(120590 gt 0)
(97)
1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin(119878)
=1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin(119878)
le 119890119873119879120590 1003817
1003817100381710038171205880
1003817100381710038171003817119871infin(119878)
(120590 lt 0)
(98)
where119873 = 1199060119909119871infin(119878)+radic12058221205902minus 1198702
2(119879)120590 minus 120582120590 and119870
2(119879)
is given in (50)
Proof Differentiating the left hand side of (95) with respectto 119905 in view of the relations (12) and (3) we obtain
119889
119889119905
120588 (119905 120593 (119905 119909)) 120593119909(119905 119909)
= [120588119905(119905 120593) + 120588
119909(119905 120593) 120593
119905(119905 119909)]
times 120593119909(119905 119909) + 120588 (119905 120593) 120593
119909119905(119905 119909)
= [120588119905(119905 120593) + 120588
119909(119905 120593) 120593
119905(119905 119909)] 120593
119909(119905 119909)
+ 120588 (119905 120593) 119906119909(119905 120593) 120593
119909(119905 119909)
= [120588119905(119905 120593) + 120588
119909(119905 120593) 120593
119905(119905 119909) + 120588 (119905 120593) 119906
119909(119905 120593)]
times 120593119909(119905 119909) = 0
(99)
This completes the proof of (95) In view of the assump-tion (96) and 120590 gt 0 we obtain 119906(119905 119909) ge minus(119872120590) (119905 119909) isin
[0 119879] times 119878By Lemma 3 and (95) we have
1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin =
1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin =
1003817100381710038171003817100381710038171003817
119890minusint119905
0
119906119909(120591sdot)119889120591
1205880(sdot)
1003817100381710038171003817100381710038171003817119871infin
le 1198901198721198791205901003817
1003817100381710038171205880(sdot)1003817100381710038171003817119871infin
(100)
To obtain (98) we use a similar argument as before Using(13) and the lower bound for 119906
119909(119905 119909) in (46) it follows that
1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin =
1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin =
1003817100381710038171003817100381710038171003817
119890minusint119905
0
119906119909(120591sdot)119889120591
1205880(sdot)
1003817100381710038171003817100381710038171003817119871infin
le 1198901198731198791205901003817
1003817100381710038171205880(sdot)1003817100381710038171003817119871infin
(101)
which proves (98) This completes the proof of Lemma 15
Proof of Theorem 12 Suppose that119879 lt infin and that (42) is notvalid Then there is some positive number119872 gt 0 such that
120590119906119909(119905 119909) ge minus119872 (119905 119909) isin [0 119879] times 119878 (102)
It now follows from Lemma 14 that |119906119909(119905 119909)| le 119862 where
119862 = 119862(119860119872 120590 1198640 120582 119906
0 119879) Therefore Theorem 11 implies
that the maximal existence time 119879 = infin which contradictswith the assumption that 119879 lt infin
Conversely the Sobolev embedding theorem 119867119904(119878) rarr
119871infin(119878) with 119904 gt 12 implies that if (70) holds the corre-
sponding solution blows up in finite time which completesthe proof of Theorem 12
Acknowledgments
The authors would like to thank the referee for commentsand suggestions This work is supported by the NationalNature Science Foundation of China (no 11171135) theNature Science Foundation of the Jiangsu Higher EducationInstitutions of China (no 09KJB110003) and the high-leveltalented person special subsidizes of Jiangsu University (no05JDG047)
References
[1] J K Hunter and R Saxton ldquoDynamics of director fieldsrdquo SIAMJournal on Applied Mathematics vol 51 no 6 pp 1498ndash15211991
[2] Z Yin ldquoOn the structure of solutions to the periodic Hunter-Saxton equationrdquo SIAM Journal on Mathematical Analysis vol36 no 1 pp 272ndash283 2004
[3] H-H Dai and M Pavlov ldquoTransformations for the Camassa-Holm equation its high-frequency limit and the Sinh-Gordonequationrdquo Journal of the Physical Society of Japan vol 67 no 11pp 3655ndash3657 1998
[4] J K Hunter and Y X Zheng ldquoOn a completely integrablenonlinear hyperbolic variational equationrdquo Physica D vol 79no 2ndash4 pp 361ndash386 1994
[5] R Camassa and D D Holm ldquoAn integrable shallow waterequation with peaked solitonsrdquo Physical Review Letters vol 71no 11 pp 1661ndash1664 1993
[6] R S Johnson ldquoCamassa-Holm Korteweg-de Vries and relatedmodels for water wavesrdquo Journal of Fluid Mechanics vol 455pp 63ndash82 2002
[7] A Constantin and B Kolev ldquoOn the geometric approach to themotion of inertialmechanical systemsrdquo Journal of Physics A vol35 no 32 pp R51ndashR79 2002
[8] P J Olver and P Rosenau ldquoTri-Hamiltonian duality betweensolitons and solitary-wave solutions having compact supportrdquoPhysical Review E vol 53 no 2 pp 1900ndash1906 1996
[9] R Beals D H Sattinger and J Szmigielski ldquoInverse scatteringsolutions of the Hunter-Saxton equationrdquo Applicable Analysisvol 78 no 3-4 pp 255ndash269 2001
[10] A Bressan and A Constantin ldquoGlobal solutions of the Hunter-Saxton equationrdquo SIAM Journal on Mathematical Analysis vol94 pp 68ndash92 2010
[11] A Bressan H Holden and X Raynaud ldquoLipschitz metric forthe Hunter-Saxton equationrdquo Journal de Mathematiques Pureset Appliquees vol 94 no 1 pp 68ndash92 2010
10 Journal of Applied Mathematics
[12] J K Hunter and Y X Zheng ldquoOn a nonlinear hyperbolicvariational equation I Global existence of weak solutionsrdquoArchive for Rational Mechanics and Analysis vol 129 no 4 pp305ndash353 1995
[13] J Lenells ldquoThe Hunter-Saxton equation describes the geodesicflow on a sphererdquo Journal of Geometry and Physics vol 57 no10 pp 2049ndash2064 2007
[14] J B Li and Y S Li ldquoBifurcations of travelling wave solutions fora two-component Camassa-Holm equationrdquoActaMathematicaSinica (English Series) vol 24 no 8 pp 1319ndash1330 2008
[15] W Rui and Y Long ldquoIntegral bifurcation method together witha translation-dilation transformation for solving an integrable2-component Camassa-Holm shallow water systemrdquo Journal ofApplied Mathematics vol 2012 Article ID 736765 21 pages2012
[16] H Wu and M Wunsch ldquoGlobal existence for the generalizedtwo-component Hunter-Saxton systemrdquo Journal of Mathemati-cal Fluid Mechanics vol 14 no 3 pp 455ndash469 2012
[17] BMoon andY Liu ldquoWave breaking and global existence for thegeneralized periodic two-component Hunter-Saxton systemrdquoJournal of Differential Equations vol 253 no 1 pp 319ndash3552012
[18] S Wu and Z Yin ldquoBlow-up blow-up rate and decay of thesolution of the weakly dissipative Camassa-Holm equationrdquoJournal ofMathematical Physics vol 47 no 1 Article ID 0135042006
[19] S Wu J Escher and Z Yin ldquoGlobal existence and blow-up phenomena for a weakly dissipative Degasperis-Procesiequationrdquo Discrete and Continuous Dynamical Systems B vol12 no 3 pp 633ndash645 2009
[20] T Kato ldquoQuasi-linear equations of evolution with applicationsto partial differential equationsrdquo in Spectral Theory and Differ-ential Equations vol 448 of Lecture Notes in Mathematics pp25ndash70 Springer Berlin Germany 1975
[21] G Gui and Y Liu ldquoOn the global existence and wave-breakingcriteria for the two-component Camassa-Holm systemrdquo Journalof Functional Analysis vol 258 no 12 pp 4251ndash4278 2010
[22] A Constantin and J Escher ldquoWave breaking for nonlinearnonlocal shallow water equationsrdquo Acta Mathematica vol 181no 2 pp 229ndash243 1998
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 5
(2) If 119891 = V then for all 119904 gt 0 estimates (35) and (36) holdwith
119881 (119905) = int
119905
0
1003817100381710038171003817120597119909119906 (120591)
1003817100381710038171003817119871infin119889120591 (37)
Proposition 10 (see [21]) Let 0 lt 119904 lt 1 Suppose that 1198910isin
119867119904 119892 isin 119871
1([0 119879]119867
119904)V 120597119909V isin 119871
1([0 119879] 119871
infin) and that 119891 isin
119871infin([0 119879]119867
119904) cap 119862([0 119879] 119878
1015840) solves the 1-dimensional linear
transport equation
(119879)
120597119905119891 + V sdot nabla119891 = 119892
1198911003816100381610038161003816119905=0
= 1198910
(38)
Then 119891 isin 119862([0 119879]H119904) More precisely there exists a con-stant 119862 depending only on 119904 such that the following statementshold
10038171003817100381710038171198911003817100381710038171003817119867119904 le
10038171003817100381710038171198910
1003817100381710038171003817119867119904 + 119862int
119905
0
1003817100381710038171003817119892 (120591)
1003817100381710038171003817119867119904119889120591
+ 119862int
119905
0
1198811015840(120591)
1003817100381710038171003817119891 (120591)
1003817100381710038171003817119867119904119889120591
(39)
or
10038171003817100381710038171198911003817100381710038171003817119867119904 le 119890119862119881(119905)
(10038171003817100381710038171198910
1003817100381710038171003817119867119904 + 119862int
119905
0
1003817100381710038171003817119892 (120591)
1003817100381710038171003817119867119904119889120591) (40)
with 119881(119905) = int
119905
0(V(120591)
119871infin + 120597
119909V(120591)119871infin)119889120591
The above proposition was proved in [8] using Little-wood-Paley analysis for the transport equation and Moser-type estimates Using this result and performing the sameargument as in [17] we can obtain the following blow-upcriterion
Theorem 11 Let1198830= (1199060
1205880
) isin 119867119904times119867119904minus1 with 119904 ge 2 and119883 =
(119906
120588 ) be the corresponding solution to (3) Assume that 119879 gt 0 isthe maximal time of existence Then
119879 lt infin 997904rArr int
119879
0
1003817100381710038171003817120597119909119906 (120591)
1003817100381710038171003817119871infin119889120591 = infin (41)
Our next result describes the necessary and sufficientcondition for the blow-up of solutions to (3)
Theorem 12 Suppose that 120590 isin 119877 0 Let 1198830= (1199060
1205880
) isin 119867119904times
119867119904minus1 with 119904 ge 2 and let119879 be themaximal existence time of the
solution119883 = (119906
120588 ) to (3)with initial data1198830 Then the solutionblows up in finite time if and only if
lim inf119905rarr119879
minus
inf119909isin119878
120590119906119909(119905 119909) = minusinfin (42)
The approach one takes here is the method of charac-teristics Applying the following lemma we may carry outthe estimates along the characteristics 120593(119905 119909) which capturessup119909isin119878
119906119909(119905 119909) and inf
119909isin119878119906119909(119905 119909)
Lemma 13 (see [22]) Let119879 gt 0 and let V isin 1198621([0 119879]1198672(119877))Then for every 119905 isin [0 119879) there exists at least one point 120585(119905) isin 119877with
119898(119905) = inf119909isin119878
V119909(119905 119909) = V
119909(119905 120585 (119905)) (43)
and the function 119898(119905) is almost everywhere differentiable on(0 119879) with
119889119898 (119905)
119889119905
= V119905119909(119905 120585 (119905)) 119886119890 119900119899 (0 119879) (44)
Lemma 14 Let 1198830= (1199060
1205880
) isin 119867119904times 119867119904minus1 with 119904 ge 2 and let 119879
be the maximal existence time of the solution 119883 = (119906
120588 ) to (3)with initial data119883
0 Then one has the following
(1) 120590 = 0
sup119909isin119878
119906119909(119905 119909) le
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
+radic
10038171003817100381710038171205880
1003817100381710038171003817
2
119871infin(119878)+ 1198702
1(119879)
120590
+
1205822
1205902+
120582
120590
(120590 gt 0)
(45)
inf119909isin119878
119906119909(119905 119909) ge minus
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
minusradic1205822
1205902minus
1198702
2(119879)
120590
+
120582
120590
(120590 lt 0)
(46)
(2) 120590 = 0
sup119909isin119878
119906119909(119905 119909) le sup
119909isin119878
1199060119909
(119909)
+
1
2
(sup119909isin119878
1205882
0(119909) + 119870
2
1(119879))
119890120582119905minus 1
120582
(47)
inf119909isin119878
119906119909(119905 119909) ge inf
119909isin119878
1199060119909
(119909)
+
1
2
(inf119909isin119878
1205882
0(119909) minus 119870
2
2(119879))
119890120582119905minus 1
120582
(48)
The constants above are defined as follows
1198701(119879) = radic2119860 minus 120582 +
119860
2
1198640+
3119860 minus 2120582
2
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(49)
1198702(119879)
= radic2119860 minus 120582 +
119860 + 2
2
1198640+
3119860 minus 2120582
2
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(50)
Proof of Lemma 14 ByTheorem 2 and a simple density argu-ment we show that the desired results are valid when 119904 ge 3so we take 119904 = 3 in the proof
6 Journal of Applied Mathematics
Let 120590 gt 0 Using Lemma 13 and the fact that
sup119909isin119878
[V119909(119905 119909)] = minusinf
119909isin119878
[minusV119909(119905 119909)] (51)
We can consider119872(119905) and 120574(119905) as follows
119872(119905) = 119906119909(119905 120585 (119905)) = sup
119909isin119878
[119906119909(119905 119909)] 119905 isin [0 119879) (52)
Hence
119906119909119909(119905 120585 (119905)) = 0 ae on 119905 isin [0 119879) (53)
Take the trajectory 120593(119905 119909) defined in (12) Then we knowthat 120593(119905 sdot) 119877 rarr 119877 is a diffeomorphism for every 119905 isin [0 119879)Therefore there exists 119909
0(119905) isin 119877 such that
120593 (119905 1199090(119905)) = 120585 (119905) 119905 isin [0 119879) (54)
Now let
120574 (119905) = 120588 (119905 120593 (119905 1199090)) 119905 isin [0 119879) (55)
Therefore along the trajectory120593(119905 1199090) (4) and the second
equation of (3) become
1198721015840(119905) = minus
120590
2
1198722(119905) + 120582119872 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
1205741015840(119905) = minus120574119872 ae 119905 isin [0 119879)
(56)
where the notation denotes the derivative with respect to 119905
and 119891 represents the function
119891 = minus119860119906 minus 120582120597minus1
119909119906 + 119892 (119905)
= minus119860119906 minus 120582120597minus1
119909119906 minus int
119878
(
120590
2
1199062
119909+
1
2
1205882minus 119860119906)119889119909
(57)
We first compute the upper and lower bounds for 119891 forlater use in getting the blow-up result as follows
119891 = minus119860119906 minus 120582int
119909
0
119906 119889119909 minus
120590
2
int
119878
1199062
119909119889119909 minus
1
2
int
119878
1205882119889119909 + 119860int
119878
119906 119889119909
le minus119860119906 minus 120582int
119909
0
119906 119889119909 + 119860int
119878
119906 119889119909 le
119860
2
(1 + 1199062)
+
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
(58)
Since 1199062 le (12) int119878(1199062+ 1199062
119909)119889119909 (17) we obtain the upper
bound for 119891
119891 le
119860
2
(1 +
1
2
int
119878
(1199062+ 1199062
119909) 119889119909) +
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le 119860 minus
120582
2
+
119860
4
int
119878
(1205882+ 1199062
119909) 119889119909 +
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860
4
1198640+
3119860 minus 2120582
4
[1198901198622119905(int
119878
1199062
0(119909) 119889119909 + 1)]
le 119860 minus
120582
2
+
119860
4
1198640+
3119860 minus 2120582
4
times [1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)] =
1
2
1198702
1(119879)
(59)
Now we turn to the lower bound of 119891 Using previous argu-ments we get
minus119891 = 119860119906 + 120582int
119909
0
119906 119889119909 +
120590
2
int
119878
1199062
119909119889119909 +
1
2
int
119878
1205882119889119909 minus 119860int
119878
119906 119889119909
le 119860
1 + 1199062
2
+
max (|120590| 1)2
int
119878
(1205882+ 1199062
119909) 119889119909
+
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le 119860 minus
120582
2
+
119860 + 2max (|120590| 1)4
int
119878
(1205882+ 1199062
119909) 119889119909
+
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860 + 2max (|120590| 1)4
1198640
+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(60)
When 120590 lt 0 we have a finer estimate
minus119891 le 119860 minus
120582
2
+
119860 + 2
4
int
119878
(1205882+ 1199062
119909) 119889119909 +
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860 + 2
4
1198640+
3119860 minus 2120582
4
times [1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)] =
1
2
1198702
2(119879)
(61)
Combining (59) and (60) we obtain
10038161003816100381610038161198911003816100381610038161003816le 119860 minus
120582
2
+
119860 + 2max (|120590| 1)4
1198640
+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)] =
1
2
1198702
3(119879)
(62)
Since 119904 ge 3 we have 119906 isin 11986210(119878) Therefore
sup119909isin119878
119906119909(119905 119909) ge 0 inf
119909isin119878
119906119909(119905 119909) le 0 119905 isin [0 119879) (63)
Journal of Applied Mathematics 7
Hence119872(119905) gt 0 for 119905 isin [0 119879) From the second equationof (55) we obtain
120574 (119905) = 120574 (0) 119890minusint119905
0
119872(120591)119889120591 (64)
Hence1003816100381610038161003816120588 (119905 120593 (119905 119909
0))1003816100381610038161003816=1003816100381610038161003816120574 (119905)
1003816100381610038161003816le1003816100381610038161003816120574 (0)
1003816100381610038161003816le10038171003817100381710038171205880
1003817100381710038171003817119871infin(119878) (65)
For any given 119909 isin 119878 define
1199011(119905) = 119872 (119905) minus
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
minusradic
10038171003817100381710038171205880
1003817100381710038171003817
2
119871infin(119878)+ 1198702
1(119879)
120590
+
1205822
1205902minus
120582
120590
(120590 gt 0)
(66)
Observing that 1199011(119905) is a 1198621-differentiable function on
[0 119879) and satisfies
1199011(0) = 119872 (0) minus
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
minusradic
10038171003817100381710038171205880
1003817100381710038171003817
2
119871infin(119878)+ 1198702
1(119879)
120590
+
1205822
1205902minus
120582
120590
le 119872(0) minus10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
(67)
We now claim that 1199011(119905) le 0 119905 isin [0 119879)
Assume the contrary that there is 1199050isin [0 119879) such that
1199011(1199050) gt 0
Let 1199051= max119905 lt 119905
0 1199011(119905) = 0 Then 119901
1(1199051) = 0 and
1199011015840
1(1199051) ge 0 or equivalently
119872(1199051) =
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)+radic
10038171003817100381710038171205880
1003817100381710038171003817
2
119871infin(119878)+ 1198702
1(119879)
120590
+
1205822
1205902+
120582
120590
(68)
and1198721015840(1199051) ge 0 ae 119905 isin [0 119879) On the other hand we have
1198721015840(1199051) = minus
120590
2
1198722(1199051) + 120582119872(119905
1) +
1
2
1205742(1199051)
+ 119891 (1199051 120593 (1199051 1199090)) ae 119905 isin [0 119879)
le minus
120590
2
(119872(1199051) minus
120582
120590
)
2
+
1
2
10038171003817100381710038171205880
1003817100381710038171003817
2
119871infin(119878)+
1205822
2120590
+
1198702
1(119879)
2
lt 0
(69)
which is a contradictionTherefore1199011(119905) le 0 for all 119905 isin [0 119879)
Since 119909 is arbitrarily chosen we obtain (45)To derive (46) in the case of 120590 lt 0 we consider (119905) and
120574(119905) as in Lemma 13
(119905) = 119906119909(119905 120577 (119905)) = inf
119909isin119878
[119906119909(119905 119909)] 119905 isin [0 119879) (70)
Hence
119906119909119909(119905 120577 (119905)) = 0 ae 119905 isin [0 119879) (71)
Using previous arguments we take the characteristic 120593(119905119909) defined in (13) and choose 119909
1(119905) isin 119877 such that
120593 (119905 1199091(119905)) = 120577 (119905) (72)
Let
120574 (119905) = 120588 (119905 120593 (119905 119909)) 119905 isin [0 119879) (73)
Hence along the trajectory 1015840(119905) = 120582(119905) +(12)1205742(119905) +
119891(119905 120593(119905 1199090)) ge 120582(0)+(12)120574
2(0) +(12)119870
2
2(119879) (4) and the
second equation of (3) become
1015840(119905) = minus
120590
2
2(119905) + 120582 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
1205741015840(119905) = minus120574 ae 119905 isin [0 119879)
(74)
Define
1199012(119905) = (119905) +
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
+radic1205822
1205902minus
1198702
2(119879)
120590
minus
120582
120590
(120590 lt 0)
(75)
For any given 119909 isin 119878 Note that 1199012(119905) is also 119862
1-dif-ferentiable function on [0 119879) and satisfies
1199012(0) = (0) +
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
+radic1205822
1205902minus
1198702
2(119879)
120590
minus
120582
120590
ge (0) +10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
ge 0
(76)
We now claim that 1199012(119905) ge 0 for any 119905 isin [0 119879)
Suppose not then there is isin [0 119879) such that 1199012(1199050) lt 0
Define
1199052= max 119905 lt 119905
0 1199012(119905) = 0 (77)
Then 1199012(1199052) = 0 and 1199011015840
2(1199052) lt 0 or equivalently
(1199052) = minus
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)minusradic1205822
1205902minus
1198702
2(119879)
120590
+
120582
120590
(78)
and 1015840(1199052) le 0 ae 119905 isin [0 119879) However we have
1015840(1199052) = minus
120590
2
2(1199052) + 120582 (119905
2) +
1
2
1205742(1199052)
+ 119891 (1199052 120593 (1199052 1199090)) ae 119905 isin [0 119879)
ge minus
120590
2
(minus10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)minusradic1205822
1205902minus
1198702
2(119879)
120590
)
2
+
1205822
2120590
minus
1
2
1198702
2(119879) gt 0
(79)
8 Journal of Applied Mathematics
Therefore 1199012(119905) ge 0 for any 119905 isin [0 119879) Since 119909 is chosen
arbitrarily we obtain (46)Let 120590 = 0 Using previous arguments (56) becomes
1198721015840(119905) = minus
120590
2
1198722(119905) + 120582119872 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
1205741015840(119905)
= minus120574119872 ae 119905 isin [0T) (80)
where the notation denotes the derivative with respect to 119905
and 119891 represents the function
119891 = minus119860119906 minus 120582120597minus1
119909119906 + 119892 (119905)
= minus119860119906 minus 120582120597minus1
119909119906 minus int
119878
(
1
2
1205882minus 119860119906)119889119909
(81)
We first compute the upper and lower bounds for 119891 forlater use in getting the blow-up result
119891 = minus119860119906 minus 120582int
119909
0
119906 119889119909 minus
1
2
int
119878
1205882119889119909 + 119860int
119878
119906 119889119909
le minus119860119906 minus 120582int
119909
0
119906 119889119909 + 119860int
119878
119906 119889119909 le
119860
2
(1 + 1199062)
+
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le
119860
2
(1 +
1
2
int
119878
(1199062+ 1199062
119909) 119889119909) +
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le 119860 minus
120582
2
+
119860
4
int
119878
(1205882+ 1199062
119909) 119889119909 +
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860
4
1198640+
3119860 minus 2120582
4
[1198901198622119905(int
119878
1199062
0(119909) 119889119909 + 1)]
le 119860 minus
120582
2
+
119860
4
1198640+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(82)
Now we turn to the lower bound of 119891
minus119891 = 119860119906 + 120582int
119909
0
119906 119889119909 +
1
2
int
119878
1205882119889119909 minus 119860int
119878
119906 119889119909
le 119860
1 + 1199062
2
+
1
2
int
119878
1205882119889119909 +
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le 119860 minus
120582
2
+
119860 + 2
4
int
119878
(1205882+ 1199062
119909) 119889119909 +
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860 + 2
4
1198640+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(83)
Combining (82) and (83) we obtain
10038161003816100381610038161198911003816100381610038161003816le 119860 minus
120582
2
+
119860 + 2
4
1198640+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(84)
We know 119872(119905) gt 0 for 119905 isin [0 119879) From the secondequation of (81) we obtain that
120574 (119905) = 120574 (0) 119890minusint119905
0
119872(120591)119889120591 (85)
Hence1003816100381610038161003816120588 (119905 120593 (119905 119909
0))1003816100381610038161003816=1003816100381610038161003816120574 (119905)
1003816100381610038161003816le1003816100381610038161003816120574 (0)
1003816100381610038161003816 (86)
Therefore we have
1198721015840(119905) = 120582119872 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
le 120582119872 (0) +
1
2
1205742(0) +
1
2
1198702
1(119879)
(87)
1198721015840(119905) minus 120582119872 (119905) le
1
2
(sup119909isin119878
1205882
0(119909) + 119870
2
1(119879)) ae 119905 isin [0 119879)
(88)
Integrating (88) on [0 119905] we prove (47) as follows
119872(119905) le sup119909isin119878
1199060119909
(119909) +
1
2
(sup119909isin119878
1205882
0(119909) + 119870
2
1(119879))
119890120582119905minus 1
120582
(89)
To obtain a lower bound for inf119909isin119878
119906119909(119905 119909) we use the
same argumentSince 120590 = 0 (80) becomes
1015840(119905) = 120582 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
1))
1205741015840(119905) = minus120574 ae 119905 isin [0 119879)
(90)
Because of (119905) lt 0 we get from the second equation of(90) that
120574 (119905) = 120574 (0) 119890minusint119905
0
119872(120591)119889120591 (91)
This means that1003816100381610038161003816120588 (119905 120593 (119905 119909
1))1003816100381610038161003816=1003816100381610038161003816120574 (119905)
1003816100381610038161003816ge1003816100381610038161003816120574 (0)
1003816100381610038161003816 (92)
Then
1015840(119905) = 120582 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
ge 120582 (0) +
1
2
1205742(0) +
1
2
1198702
2(119879)
(93)
1015840(119905) minus 120582 (119905) ge
1
2
(inf119909isin119878
1205882
0(119909) minus 119870
2
2(119879)) ae 119905 isin [0 119879)
(94)
Integrating (94) on [0 119905] we prove (48) This completesthe proof of Lemma 14
Lemma 15 Suppose that 120590 isin 119877 0 Suppose 1198830= (1199060
1205880
) isin
119867119904times119867119904minus1 with 119904 ge 2 and let 119879 be the maximal existence time
Journal of Applied Mathematics 9
of the solution 119883 = (119906
120588 ) to (3) with initial data 1198830 Then we
have
120588 (119905 120593 (119905 119909)) 120593119909(119905 119909) = 120588
0(119909) (119905 119909) isin [0 119879] times 119878 (95)
Moreover if there exists119872 gt 0 such that
inf(119905119909)isin[0119879]times119878
120590119906119909(119905 119909) ge minus119872 (119905 119909) isin [0 119879] times 119878 (96)
Then1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin(119878)
=1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin(119878)
le 119890119872119879120590 1003817
1003817100381710038171205880
1003817100381710038171003817119871infin(119878)
(120590 gt 0)
(97)
1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin(119878)
=1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin(119878)
le 119890119873119879120590 1003817
1003817100381710038171205880
1003817100381710038171003817119871infin(119878)
(120590 lt 0)
(98)
where119873 = 1199060119909119871infin(119878)+radic12058221205902minus 1198702
2(119879)120590 minus 120582120590 and119870
2(119879)
is given in (50)
Proof Differentiating the left hand side of (95) with respectto 119905 in view of the relations (12) and (3) we obtain
119889
119889119905
120588 (119905 120593 (119905 119909)) 120593119909(119905 119909)
= [120588119905(119905 120593) + 120588
119909(119905 120593) 120593
119905(119905 119909)]
times 120593119909(119905 119909) + 120588 (119905 120593) 120593
119909119905(119905 119909)
= [120588119905(119905 120593) + 120588
119909(119905 120593) 120593
119905(119905 119909)] 120593
119909(119905 119909)
+ 120588 (119905 120593) 119906119909(119905 120593) 120593
119909(119905 119909)
= [120588119905(119905 120593) + 120588
119909(119905 120593) 120593
119905(119905 119909) + 120588 (119905 120593) 119906
119909(119905 120593)]
times 120593119909(119905 119909) = 0
(99)
This completes the proof of (95) In view of the assump-tion (96) and 120590 gt 0 we obtain 119906(119905 119909) ge minus(119872120590) (119905 119909) isin
[0 119879] times 119878By Lemma 3 and (95) we have
1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin =
1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin =
1003817100381710038171003817100381710038171003817
119890minusint119905
0
119906119909(120591sdot)119889120591
1205880(sdot)
1003817100381710038171003817100381710038171003817119871infin
le 1198901198721198791205901003817
1003817100381710038171205880(sdot)1003817100381710038171003817119871infin
(100)
To obtain (98) we use a similar argument as before Using(13) and the lower bound for 119906
119909(119905 119909) in (46) it follows that
1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin =
1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin =
1003817100381710038171003817100381710038171003817
119890minusint119905
0
119906119909(120591sdot)119889120591
1205880(sdot)
1003817100381710038171003817100381710038171003817119871infin
le 1198901198731198791205901003817
1003817100381710038171205880(sdot)1003817100381710038171003817119871infin
(101)
which proves (98) This completes the proof of Lemma 15
Proof of Theorem 12 Suppose that119879 lt infin and that (42) is notvalid Then there is some positive number119872 gt 0 such that
120590119906119909(119905 119909) ge minus119872 (119905 119909) isin [0 119879] times 119878 (102)
It now follows from Lemma 14 that |119906119909(119905 119909)| le 119862 where
119862 = 119862(119860119872 120590 1198640 120582 119906
0 119879) Therefore Theorem 11 implies
that the maximal existence time 119879 = infin which contradictswith the assumption that 119879 lt infin
Conversely the Sobolev embedding theorem 119867119904(119878) rarr
119871infin(119878) with 119904 gt 12 implies that if (70) holds the corre-
sponding solution blows up in finite time which completesthe proof of Theorem 12
Acknowledgments
The authors would like to thank the referee for commentsand suggestions This work is supported by the NationalNature Science Foundation of China (no 11171135) theNature Science Foundation of the Jiangsu Higher EducationInstitutions of China (no 09KJB110003) and the high-leveltalented person special subsidizes of Jiangsu University (no05JDG047)
References
[1] J K Hunter and R Saxton ldquoDynamics of director fieldsrdquo SIAMJournal on Applied Mathematics vol 51 no 6 pp 1498ndash15211991
[2] Z Yin ldquoOn the structure of solutions to the periodic Hunter-Saxton equationrdquo SIAM Journal on Mathematical Analysis vol36 no 1 pp 272ndash283 2004
[3] H-H Dai and M Pavlov ldquoTransformations for the Camassa-Holm equation its high-frequency limit and the Sinh-Gordonequationrdquo Journal of the Physical Society of Japan vol 67 no 11pp 3655ndash3657 1998
[4] J K Hunter and Y X Zheng ldquoOn a completely integrablenonlinear hyperbolic variational equationrdquo Physica D vol 79no 2ndash4 pp 361ndash386 1994
[5] R Camassa and D D Holm ldquoAn integrable shallow waterequation with peaked solitonsrdquo Physical Review Letters vol 71no 11 pp 1661ndash1664 1993
[6] R S Johnson ldquoCamassa-Holm Korteweg-de Vries and relatedmodels for water wavesrdquo Journal of Fluid Mechanics vol 455pp 63ndash82 2002
[7] A Constantin and B Kolev ldquoOn the geometric approach to themotion of inertialmechanical systemsrdquo Journal of Physics A vol35 no 32 pp R51ndashR79 2002
[8] P J Olver and P Rosenau ldquoTri-Hamiltonian duality betweensolitons and solitary-wave solutions having compact supportrdquoPhysical Review E vol 53 no 2 pp 1900ndash1906 1996
[9] R Beals D H Sattinger and J Szmigielski ldquoInverse scatteringsolutions of the Hunter-Saxton equationrdquo Applicable Analysisvol 78 no 3-4 pp 255ndash269 2001
[10] A Bressan and A Constantin ldquoGlobal solutions of the Hunter-Saxton equationrdquo SIAM Journal on Mathematical Analysis vol94 pp 68ndash92 2010
[11] A Bressan H Holden and X Raynaud ldquoLipschitz metric forthe Hunter-Saxton equationrdquo Journal de Mathematiques Pureset Appliquees vol 94 no 1 pp 68ndash92 2010
10 Journal of Applied Mathematics
[12] J K Hunter and Y X Zheng ldquoOn a nonlinear hyperbolicvariational equation I Global existence of weak solutionsrdquoArchive for Rational Mechanics and Analysis vol 129 no 4 pp305ndash353 1995
[13] J Lenells ldquoThe Hunter-Saxton equation describes the geodesicflow on a sphererdquo Journal of Geometry and Physics vol 57 no10 pp 2049ndash2064 2007
[14] J B Li and Y S Li ldquoBifurcations of travelling wave solutions fora two-component Camassa-Holm equationrdquoActaMathematicaSinica (English Series) vol 24 no 8 pp 1319ndash1330 2008
[15] W Rui and Y Long ldquoIntegral bifurcation method together witha translation-dilation transformation for solving an integrable2-component Camassa-Holm shallow water systemrdquo Journal ofApplied Mathematics vol 2012 Article ID 736765 21 pages2012
[16] H Wu and M Wunsch ldquoGlobal existence for the generalizedtwo-component Hunter-Saxton systemrdquo Journal of Mathemati-cal Fluid Mechanics vol 14 no 3 pp 455ndash469 2012
[17] BMoon andY Liu ldquoWave breaking and global existence for thegeneralized periodic two-component Hunter-Saxton systemrdquoJournal of Differential Equations vol 253 no 1 pp 319ndash3552012
[18] S Wu and Z Yin ldquoBlow-up blow-up rate and decay of thesolution of the weakly dissipative Camassa-Holm equationrdquoJournal ofMathematical Physics vol 47 no 1 Article ID 0135042006
[19] S Wu J Escher and Z Yin ldquoGlobal existence and blow-up phenomena for a weakly dissipative Degasperis-Procesiequationrdquo Discrete and Continuous Dynamical Systems B vol12 no 3 pp 633ndash645 2009
[20] T Kato ldquoQuasi-linear equations of evolution with applicationsto partial differential equationsrdquo in Spectral Theory and Differ-ential Equations vol 448 of Lecture Notes in Mathematics pp25ndash70 Springer Berlin Germany 1975
[21] G Gui and Y Liu ldquoOn the global existence and wave-breakingcriteria for the two-component Camassa-Holm systemrdquo Journalof Functional Analysis vol 258 no 12 pp 4251ndash4278 2010
[22] A Constantin and J Escher ldquoWave breaking for nonlinearnonlocal shallow water equationsrdquo Acta Mathematica vol 181no 2 pp 229ndash243 1998
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Applied Mathematics
Let 120590 gt 0 Using Lemma 13 and the fact that
sup119909isin119878
[V119909(119905 119909)] = minusinf
119909isin119878
[minusV119909(119905 119909)] (51)
We can consider119872(119905) and 120574(119905) as follows
119872(119905) = 119906119909(119905 120585 (119905)) = sup
119909isin119878
[119906119909(119905 119909)] 119905 isin [0 119879) (52)
Hence
119906119909119909(119905 120585 (119905)) = 0 ae on 119905 isin [0 119879) (53)
Take the trajectory 120593(119905 119909) defined in (12) Then we knowthat 120593(119905 sdot) 119877 rarr 119877 is a diffeomorphism for every 119905 isin [0 119879)Therefore there exists 119909
0(119905) isin 119877 such that
120593 (119905 1199090(119905)) = 120585 (119905) 119905 isin [0 119879) (54)
Now let
120574 (119905) = 120588 (119905 120593 (119905 1199090)) 119905 isin [0 119879) (55)
Therefore along the trajectory120593(119905 1199090) (4) and the second
equation of (3) become
1198721015840(119905) = minus
120590
2
1198722(119905) + 120582119872 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
1205741015840(119905) = minus120574119872 ae 119905 isin [0 119879)
(56)
where the notation denotes the derivative with respect to 119905
and 119891 represents the function
119891 = minus119860119906 minus 120582120597minus1
119909119906 + 119892 (119905)
= minus119860119906 minus 120582120597minus1
119909119906 minus int
119878
(
120590
2
1199062
119909+
1
2
1205882minus 119860119906)119889119909
(57)
We first compute the upper and lower bounds for 119891 forlater use in getting the blow-up result as follows
119891 = minus119860119906 minus 120582int
119909
0
119906 119889119909 minus
120590
2
int
119878
1199062
119909119889119909 minus
1
2
int
119878
1205882119889119909 + 119860int
119878
119906 119889119909
le minus119860119906 minus 120582int
119909
0
119906 119889119909 + 119860int
119878
119906 119889119909 le
119860
2
(1 + 1199062)
+
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
(58)
Since 1199062 le (12) int119878(1199062+ 1199062
119909)119889119909 (17) we obtain the upper
bound for 119891
119891 le
119860
2
(1 +
1
2
int
119878
(1199062+ 1199062
119909) 119889119909) +
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le 119860 minus
120582
2
+
119860
4
int
119878
(1205882+ 1199062
119909) 119889119909 +
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860
4
1198640+
3119860 minus 2120582
4
[1198901198622119905(int
119878
1199062
0(119909) 119889119909 + 1)]
le 119860 minus
120582
2
+
119860
4
1198640+
3119860 minus 2120582
4
times [1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)] =
1
2
1198702
1(119879)
(59)
Now we turn to the lower bound of 119891 Using previous argu-ments we get
minus119891 = 119860119906 + 120582int
119909
0
119906 119889119909 +
120590
2
int
119878
1199062
119909119889119909 +
1
2
int
119878
1205882119889119909 minus 119860int
119878
119906 119889119909
le 119860
1 + 1199062
2
+
max (|120590| 1)2
int
119878
(1205882+ 1199062
119909) 119889119909
+
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le 119860 minus
120582
2
+
119860 + 2max (|120590| 1)4
int
119878
(1205882+ 1199062
119909) 119889119909
+
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860 + 2max (|120590| 1)4
1198640
+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(60)
When 120590 lt 0 we have a finer estimate
minus119891 le 119860 minus
120582
2
+
119860 + 2
4
int
119878
(1205882+ 1199062
119909) 119889119909 +
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860 + 2
4
1198640+
3119860 minus 2120582
4
times [1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)] =
1
2
1198702
2(119879)
(61)
Combining (59) and (60) we obtain
10038161003816100381610038161198911003816100381610038161003816le 119860 minus
120582
2
+
119860 + 2max (|120590| 1)4
1198640
+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)] =
1
2
1198702
3(119879)
(62)
Since 119904 ge 3 we have 119906 isin 11986210(119878) Therefore
sup119909isin119878
119906119909(119905 119909) ge 0 inf
119909isin119878
119906119909(119905 119909) le 0 119905 isin [0 119879) (63)
Journal of Applied Mathematics 7
Hence119872(119905) gt 0 for 119905 isin [0 119879) From the second equationof (55) we obtain
120574 (119905) = 120574 (0) 119890minusint119905
0
119872(120591)119889120591 (64)
Hence1003816100381610038161003816120588 (119905 120593 (119905 119909
0))1003816100381610038161003816=1003816100381610038161003816120574 (119905)
1003816100381610038161003816le1003816100381610038161003816120574 (0)
1003816100381610038161003816le10038171003817100381710038171205880
1003817100381710038171003817119871infin(119878) (65)
For any given 119909 isin 119878 define
1199011(119905) = 119872 (119905) minus
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
minusradic
10038171003817100381710038171205880
1003817100381710038171003817
2
119871infin(119878)+ 1198702
1(119879)
120590
+
1205822
1205902minus
120582
120590
(120590 gt 0)
(66)
Observing that 1199011(119905) is a 1198621-differentiable function on
[0 119879) and satisfies
1199011(0) = 119872 (0) minus
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
minusradic
10038171003817100381710038171205880
1003817100381710038171003817
2
119871infin(119878)+ 1198702
1(119879)
120590
+
1205822
1205902minus
120582
120590
le 119872(0) minus10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
(67)
We now claim that 1199011(119905) le 0 119905 isin [0 119879)
Assume the contrary that there is 1199050isin [0 119879) such that
1199011(1199050) gt 0
Let 1199051= max119905 lt 119905
0 1199011(119905) = 0 Then 119901
1(1199051) = 0 and
1199011015840
1(1199051) ge 0 or equivalently
119872(1199051) =
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)+radic
10038171003817100381710038171205880
1003817100381710038171003817
2
119871infin(119878)+ 1198702
1(119879)
120590
+
1205822
1205902+
120582
120590
(68)
and1198721015840(1199051) ge 0 ae 119905 isin [0 119879) On the other hand we have
1198721015840(1199051) = minus
120590
2
1198722(1199051) + 120582119872(119905
1) +
1
2
1205742(1199051)
+ 119891 (1199051 120593 (1199051 1199090)) ae 119905 isin [0 119879)
le minus
120590
2
(119872(1199051) minus
120582
120590
)
2
+
1
2
10038171003817100381710038171205880
1003817100381710038171003817
2
119871infin(119878)+
1205822
2120590
+
1198702
1(119879)
2
lt 0
(69)
which is a contradictionTherefore1199011(119905) le 0 for all 119905 isin [0 119879)
Since 119909 is arbitrarily chosen we obtain (45)To derive (46) in the case of 120590 lt 0 we consider (119905) and
120574(119905) as in Lemma 13
(119905) = 119906119909(119905 120577 (119905)) = inf
119909isin119878
[119906119909(119905 119909)] 119905 isin [0 119879) (70)
Hence
119906119909119909(119905 120577 (119905)) = 0 ae 119905 isin [0 119879) (71)
Using previous arguments we take the characteristic 120593(119905119909) defined in (13) and choose 119909
1(119905) isin 119877 such that
120593 (119905 1199091(119905)) = 120577 (119905) (72)
Let
120574 (119905) = 120588 (119905 120593 (119905 119909)) 119905 isin [0 119879) (73)
Hence along the trajectory 1015840(119905) = 120582(119905) +(12)1205742(119905) +
119891(119905 120593(119905 1199090)) ge 120582(0)+(12)120574
2(0) +(12)119870
2
2(119879) (4) and the
second equation of (3) become
1015840(119905) = minus
120590
2
2(119905) + 120582 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
1205741015840(119905) = minus120574 ae 119905 isin [0 119879)
(74)
Define
1199012(119905) = (119905) +
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
+radic1205822
1205902minus
1198702
2(119879)
120590
minus
120582
120590
(120590 lt 0)
(75)
For any given 119909 isin 119878 Note that 1199012(119905) is also 119862
1-dif-ferentiable function on [0 119879) and satisfies
1199012(0) = (0) +
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
+radic1205822
1205902minus
1198702
2(119879)
120590
minus
120582
120590
ge (0) +10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
ge 0
(76)
We now claim that 1199012(119905) ge 0 for any 119905 isin [0 119879)
Suppose not then there is isin [0 119879) such that 1199012(1199050) lt 0
Define
1199052= max 119905 lt 119905
0 1199012(119905) = 0 (77)
Then 1199012(1199052) = 0 and 1199011015840
2(1199052) lt 0 or equivalently
(1199052) = minus
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)minusradic1205822
1205902minus
1198702
2(119879)
120590
+
120582
120590
(78)
and 1015840(1199052) le 0 ae 119905 isin [0 119879) However we have
1015840(1199052) = minus
120590
2
2(1199052) + 120582 (119905
2) +
1
2
1205742(1199052)
+ 119891 (1199052 120593 (1199052 1199090)) ae 119905 isin [0 119879)
ge minus
120590
2
(minus10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)minusradic1205822
1205902minus
1198702
2(119879)
120590
)
2
+
1205822
2120590
minus
1
2
1198702
2(119879) gt 0
(79)
8 Journal of Applied Mathematics
Therefore 1199012(119905) ge 0 for any 119905 isin [0 119879) Since 119909 is chosen
arbitrarily we obtain (46)Let 120590 = 0 Using previous arguments (56) becomes
1198721015840(119905) = minus
120590
2
1198722(119905) + 120582119872 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
1205741015840(119905)
= minus120574119872 ae 119905 isin [0T) (80)
where the notation denotes the derivative with respect to 119905
and 119891 represents the function
119891 = minus119860119906 minus 120582120597minus1
119909119906 + 119892 (119905)
= minus119860119906 minus 120582120597minus1
119909119906 minus int
119878
(
1
2
1205882minus 119860119906)119889119909
(81)
We first compute the upper and lower bounds for 119891 forlater use in getting the blow-up result
119891 = minus119860119906 minus 120582int
119909
0
119906 119889119909 minus
1
2
int
119878
1205882119889119909 + 119860int
119878
119906 119889119909
le minus119860119906 minus 120582int
119909
0
119906 119889119909 + 119860int
119878
119906 119889119909 le
119860
2
(1 + 1199062)
+
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le
119860
2
(1 +
1
2
int
119878
(1199062+ 1199062
119909) 119889119909) +
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le 119860 minus
120582
2
+
119860
4
int
119878
(1205882+ 1199062
119909) 119889119909 +
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860
4
1198640+
3119860 minus 2120582
4
[1198901198622119905(int
119878
1199062
0(119909) 119889119909 + 1)]
le 119860 minus
120582
2
+
119860
4
1198640+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(82)
Now we turn to the lower bound of 119891
minus119891 = 119860119906 + 120582int
119909
0
119906 119889119909 +
1
2
int
119878
1205882119889119909 minus 119860int
119878
119906 119889119909
le 119860
1 + 1199062
2
+
1
2
int
119878
1205882119889119909 +
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le 119860 minus
120582
2
+
119860 + 2
4
int
119878
(1205882+ 1199062
119909) 119889119909 +
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860 + 2
4
1198640+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(83)
Combining (82) and (83) we obtain
10038161003816100381610038161198911003816100381610038161003816le 119860 minus
120582
2
+
119860 + 2
4
1198640+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(84)
We know 119872(119905) gt 0 for 119905 isin [0 119879) From the secondequation of (81) we obtain that
120574 (119905) = 120574 (0) 119890minusint119905
0
119872(120591)119889120591 (85)
Hence1003816100381610038161003816120588 (119905 120593 (119905 119909
0))1003816100381610038161003816=1003816100381610038161003816120574 (119905)
1003816100381610038161003816le1003816100381610038161003816120574 (0)
1003816100381610038161003816 (86)
Therefore we have
1198721015840(119905) = 120582119872 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
le 120582119872 (0) +
1
2
1205742(0) +
1
2
1198702
1(119879)
(87)
1198721015840(119905) minus 120582119872 (119905) le
1
2
(sup119909isin119878
1205882
0(119909) + 119870
2
1(119879)) ae 119905 isin [0 119879)
(88)
Integrating (88) on [0 119905] we prove (47) as follows
119872(119905) le sup119909isin119878
1199060119909
(119909) +
1
2
(sup119909isin119878
1205882
0(119909) + 119870
2
1(119879))
119890120582119905minus 1
120582
(89)
To obtain a lower bound for inf119909isin119878
119906119909(119905 119909) we use the
same argumentSince 120590 = 0 (80) becomes
1015840(119905) = 120582 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
1))
1205741015840(119905) = minus120574 ae 119905 isin [0 119879)
(90)
Because of (119905) lt 0 we get from the second equation of(90) that
120574 (119905) = 120574 (0) 119890minusint119905
0
119872(120591)119889120591 (91)
This means that1003816100381610038161003816120588 (119905 120593 (119905 119909
1))1003816100381610038161003816=1003816100381610038161003816120574 (119905)
1003816100381610038161003816ge1003816100381610038161003816120574 (0)
1003816100381610038161003816 (92)
Then
1015840(119905) = 120582 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
ge 120582 (0) +
1
2
1205742(0) +
1
2
1198702
2(119879)
(93)
1015840(119905) minus 120582 (119905) ge
1
2
(inf119909isin119878
1205882
0(119909) minus 119870
2
2(119879)) ae 119905 isin [0 119879)
(94)
Integrating (94) on [0 119905] we prove (48) This completesthe proof of Lemma 14
Lemma 15 Suppose that 120590 isin 119877 0 Suppose 1198830= (1199060
1205880
) isin
119867119904times119867119904minus1 with 119904 ge 2 and let 119879 be the maximal existence time
Journal of Applied Mathematics 9
of the solution 119883 = (119906
120588 ) to (3) with initial data 1198830 Then we
have
120588 (119905 120593 (119905 119909)) 120593119909(119905 119909) = 120588
0(119909) (119905 119909) isin [0 119879] times 119878 (95)
Moreover if there exists119872 gt 0 such that
inf(119905119909)isin[0119879]times119878
120590119906119909(119905 119909) ge minus119872 (119905 119909) isin [0 119879] times 119878 (96)
Then1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin(119878)
=1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin(119878)
le 119890119872119879120590 1003817
1003817100381710038171205880
1003817100381710038171003817119871infin(119878)
(120590 gt 0)
(97)
1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin(119878)
=1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin(119878)
le 119890119873119879120590 1003817
1003817100381710038171205880
1003817100381710038171003817119871infin(119878)
(120590 lt 0)
(98)
where119873 = 1199060119909119871infin(119878)+radic12058221205902minus 1198702
2(119879)120590 minus 120582120590 and119870
2(119879)
is given in (50)
Proof Differentiating the left hand side of (95) with respectto 119905 in view of the relations (12) and (3) we obtain
119889
119889119905
120588 (119905 120593 (119905 119909)) 120593119909(119905 119909)
= [120588119905(119905 120593) + 120588
119909(119905 120593) 120593
119905(119905 119909)]
times 120593119909(119905 119909) + 120588 (119905 120593) 120593
119909119905(119905 119909)
= [120588119905(119905 120593) + 120588
119909(119905 120593) 120593
119905(119905 119909)] 120593
119909(119905 119909)
+ 120588 (119905 120593) 119906119909(119905 120593) 120593
119909(119905 119909)
= [120588119905(119905 120593) + 120588
119909(119905 120593) 120593
119905(119905 119909) + 120588 (119905 120593) 119906
119909(119905 120593)]
times 120593119909(119905 119909) = 0
(99)
This completes the proof of (95) In view of the assump-tion (96) and 120590 gt 0 we obtain 119906(119905 119909) ge minus(119872120590) (119905 119909) isin
[0 119879] times 119878By Lemma 3 and (95) we have
1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin =
1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin =
1003817100381710038171003817100381710038171003817
119890minusint119905
0
119906119909(120591sdot)119889120591
1205880(sdot)
1003817100381710038171003817100381710038171003817119871infin
le 1198901198721198791205901003817
1003817100381710038171205880(sdot)1003817100381710038171003817119871infin
(100)
To obtain (98) we use a similar argument as before Using(13) and the lower bound for 119906
119909(119905 119909) in (46) it follows that
1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin =
1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin =
1003817100381710038171003817100381710038171003817
119890minusint119905
0
119906119909(120591sdot)119889120591
1205880(sdot)
1003817100381710038171003817100381710038171003817119871infin
le 1198901198731198791205901003817
1003817100381710038171205880(sdot)1003817100381710038171003817119871infin
(101)
which proves (98) This completes the proof of Lemma 15
Proof of Theorem 12 Suppose that119879 lt infin and that (42) is notvalid Then there is some positive number119872 gt 0 such that
120590119906119909(119905 119909) ge minus119872 (119905 119909) isin [0 119879] times 119878 (102)
It now follows from Lemma 14 that |119906119909(119905 119909)| le 119862 where
119862 = 119862(119860119872 120590 1198640 120582 119906
0 119879) Therefore Theorem 11 implies
that the maximal existence time 119879 = infin which contradictswith the assumption that 119879 lt infin
Conversely the Sobolev embedding theorem 119867119904(119878) rarr
119871infin(119878) with 119904 gt 12 implies that if (70) holds the corre-
sponding solution blows up in finite time which completesthe proof of Theorem 12
Acknowledgments
The authors would like to thank the referee for commentsand suggestions This work is supported by the NationalNature Science Foundation of China (no 11171135) theNature Science Foundation of the Jiangsu Higher EducationInstitutions of China (no 09KJB110003) and the high-leveltalented person special subsidizes of Jiangsu University (no05JDG047)
References
[1] J K Hunter and R Saxton ldquoDynamics of director fieldsrdquo SIAMJournal on Applied Mathematics vol 51 no 6 pp 1498ndash15211991
[2] Z Yin ldquoOn the structure of solutions to the periodic Hunter-Saxton equationrdquo SIAM Journal on Mathematical Analysis vol36 no 1 pp 272ndash283 2004
[3] H-H Dai and M Pavlov ldquoTransformations for the Camassa-Holm equation its high-frequency limit and the Sinh-Gordonequationrdquo Journal of the Physical Society of Japan vol 67 no 11pp 3655ndash3657 1998
[4] J K Hunter and Y X Zheng ldquoOn a completely integrablenonlinear hyperbolic variational equationrdquo Physica D vol 79no 2ndash4 pp 361ndash386 1994
[5] R Camassa and D D Holm ldquoAn integrable shallow waterequation with peaked solitonsrdquo Physical Review Letters vol 71no 11 pp 1661ndash1664 1993
[6] R S Johnson ldquoCamassa-Holm Korteweg-de Vries and relatedmodels for water wavesrdquo Journal of Fluid Mechanics vol 455pp 63ndash82 2002
[7] A Constantin and B Kolev ldquoOn the geometric approach to themotion of inertialmechanical systemsrdquo Journal of Physics A vol35 no 32 pp R51ndashR79 2002
[8] P J Olver and P Rosenau ldquoTri-Hamiltonian duality betweensolitons and solitary-wave solutions having compact supportrdquoPhysical Review E vol 53 no 2 pp 1900ndash1906 1996
[9] R Beals D H Sattinger and J Szmigielski ldquoInverse scatteringsolutions of the Hunter-Saxton equationrdquo Applicable Analysisvol 78 no 3-4 pp 255ndash269 2001
[10] A Bressan and A Constantin ldquoGlobal solutions of the Hunter-Saxton equationrdquo SIAM Journal on Mathematical Analysis vol94 pp 68ndash92 2010
[11] A Bressan H Holden and X Raynaud ldquoLipschitz metric forthe Hunter-Saxton equationrdquo Journal de Mathematiques Pureset Appliquees vol 94 no 1 pp 68ndash92 2010
10 Journal of Applied Mathematics
[12] J K Hunter and Y X Zheng ldquoOn a nonlinear hyperbolicvariational equation I Global existence of weak solutionsrdquoArchive for Rational Mechanics and Analysis vol 129 no 4 pp305ndash353 1995
[13] J Lenells ldquoThe Hunter-Saxton equation describes the geodesicflow on a sphererdquo Journal of Geometry and Physics vol 57 no10 pp 2049ndash2064 2007
[14] J B Li and Y S Li ldquoBifurcations of travelling wave solutions fora two-component Camassa-Holm equationrdquoActaMathematicaSinica (English Series) vol 24 no 8 pp 1319ndash1330 2008
[15] W Rui and Y Long ldquoIntegral bifurcation method together witha translation-dilation transformation for solving an integrable2-component Camassa-Holm shallow water systemrdquo Journal ofApplied Mathematics vol 2012 Article ID 736765 21 pages2012
[16] H Wu and M Wunsch ldquoGlobal existence for the generalizedtwo-component Hunter-Saxton systemrdquo Journal of Mathemati-cal Fluid Mechanics vol 14 no 3 pp 455ndash469 2012
[17] BMoon andY Liu ldquoWave breaking and global existence for thegeneralized periodic two-component Hunter-Saxton systemrdquoJournal of Differential Equations vol 253 no 1 pp 319ndash3552012
[18] S Wu and Z Yin ldquoBlow-up blow-up rate and decay of thesolution of the weakly dissipative Camassa-Holm equationrdquoJournal ofMathematical Physics vol 47 no 1 Article ID 0135042006
[19] S Wu J Escher and Z Yin ldquoGlobal existence and blow-up phenomena for a weakly dissipative Degasperis-Procesiequationrdquo Discrete and Continuous Dynamical Systems B vol12 no 3 pp 633ndash645 2009
[20] T Kato ldquoQuasi-linear equations of evolution with applicationsto partial differential equationsrdquo in Spectral Theory and Differ-ential Equations vol 448 of Lecture Notes in Mathematics pp25ndash70 Springer Berlin Germany 1975
[21] G Gui and Y Liu ldquoOn the global existence and wave-breakingcriteria for the two-component Camassa-Holm systemrdquo Journalof Functional Analysis vol 258 no 12 pp 4251ndash4278 2010
[22] A Constantin and J Escher ldquoWave breaking for nonlinearnonlocal shallow water equationsrdquo Acta Mathematica vol 181no 2 pp 229ndash243 1998
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 7
Hence119872(119905) gt 0 for 119905 isin [0 119879) From the second equationof (55) we obtain
120574 (119905) = 120574 (0) 119890minusint119905
0
119872(120591)119889120591 (64)
Hence1003816100381610038161003816120588 (119905 120593 (119905 119909
0))1003816100381610038161003816=1003816100381610038161003816120574 (119905)
1003816100381610038161003816le1003816100381610038161003816120574 (0)
1003816100381610038161003816le10038171003817100381710038171205880
1003817100381710038171003817119871infin(119878) (65)
For any given 119909 isin 119878 define
1199011(119905) = 119872 (119905) minus
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
minusradic
10038171003817100381710038171205880
1003817100381710038171003817
2
119871infin(119878)+ 1198702
1(119879)
120590
+
1205822
1205902minus
120582
120590
(120590 gt 0)
(66)
Observing that 1199011(119905) is a 1198621-differentiable function on
[0 119879) and satisfies
1199011(0) = 119872 (0) minus
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
minusradic
10038171003817100381710038171205880
1003817100381710038171003817
2
119871infin(119878)+ 1198702
1(119879)
120590
+
1205822
1205902minus
120582
120590
le 119872(0) minus10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
(67)
We now claim that 1199011(119905) le 0 119905 isin [0 119879)
Assume the contrary that there is 1199050isin [0 119879) such that
1199011(1199050) gt 0
Let 1199051= max119905 lt 119905
0 1199011(119905) = 0 Then 119901
1(1199051) = 0 and
1199011015840
1(1199051) ge 0 or equivalently
119872(1199051) =
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)+radic
10038171003817100381710038171205880
1003817100381710038171003817
2
119871infin(119878)+ 1198702
1(119879)
120590
+
1205822
1205902+
120582
120590
(68)
and1198721015840(1199051) ge 0 ae 119905 isin [0 119879) On the other hand we have
1198721015840(1199051) = minus
120590
2
1198722(1199051) + 120582119872(119905
1) +
1
2
1205742(1199051)
+ 119891 (1199051 120593 (1199051 1199090)) ae 119905 isin [0 119879)
le minus
120590
2
(119872(1199051) minus
120582
120590
)
2
+
1
2
10038171003817100381710038171205880
1003817100381710038171003817
2
119871infin(119878)+
1205822
2120590
+
1198702
1(119879)
2
lt 0
(69)
which is a contradictionTherefore1199011(119905) le 0 for all 119905 isin [0 119879)
Since 119909 is arbitrarily chosen we obtain (45)To derive (46) in the case of 120590 lt 0 we consider (119905) and
120574(119905) as in Lemma 13
(119905) = 119906119909(119905 120577 (119905)) = inf
119909isin119878
[119906119909(119905 119909)] 119905 isin [0 119879) (70)
Hence
119906119909119909(119905 120577 (119905)) = 0 ae 119905 isin [0 119879) (71)
Using previous arguments we take the characteristic 120593(119905119909) defined in (13) and choose 119909
1(119905) isin 119877 such that
120593 (119905 1199091(119905)) = 120577 (119905) (72)
Let
120574 (119905) = 120588 (119905 120593 (119905 119909)) 119905 isin [0 119879) (73)
Hence along the trajectory 1015840(119905) = 120582(119905) +(12)1205742(119905) +
119891(119905 120593(119905 1199090)) ge 120582(0)+(12)120574
2(0) +(12)119870
2
2(119879) (4) and the
second equation of (3) become
1015840(119905) = minus
120590
2
2(119905) + 120582 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
1205741015840(119905) = minus120574 ae 119905 isin [0 119879)
(74)
Define
1199012(119905) = (119905) +
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
+radic1205822
1205902minus
1198702
2(119879)
120590
minus
120582
120590
(120590 lt 0)
(75)
For any given 119909 isin 119878 Note that 1199012(119905) is also 119862
1-dif-ferentiable function on [0 119879) and satisfies
1199012(0) = (0) +
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
+radic1205822
1205902minus
1198702
2(119879)
120590
minus
120582
120590
ge (0) +10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)
ge 0
(76)
We now claim that 1199012(119905) ge 0 for any 119905 isin [0 119879)
Suppose not then there is isin [0 119879) such that 1199012(1199050) lt 0
Define
1199052= max 119905 lt 119905
0 1199012(119905) = 0 (77)
Then 1199012(1199052) = 0 and 1199011015840
2(1199052) lt 0 or equivalently
(1199052) = minus
10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)minusradic1205822
1205902minus
1198702
2(119879)
120590
+
120582
120590
(78)
and 1015840(1199052) le 0 ae 119905 isin [0 119879) However we have
1015840(1199052) = minus
120590
2
2(1199052) + 120582 (119905
2) +
1
2
1205742(1199052)
+ 119891 (1199052 120593 (1199052 1199090)) ae 119905 isin [0 119879)
ge minus
120590
2
(minus10038171003817100381710038171199060119909
1003817100381710038171003817119871infin(119878)minusradic1205822
1205902minus
1198702
2(119879)
120590
)
2
+
1205822
2120590
minus
1
2
1198702
2(119879) gt 0
(79)
8 Journal of Applied Mathematics
Therefore 1199012(119905) ge 0 for any 119905 isin [0 119879) Since 119909 is chosen
arbitrarily we obtain (46)Let 120590 = 0 Using previous arguments (56) becomes
1198721015840(119905) = minus
120590
2
1198722(119905) + 120582119872 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
1205741015840(119905)
= minus120574119872 ae 119905 isin [0T) (80)
where the notation denotes the derivative with respect to 119905
and 119891 represents the function
119891 = minus119860119906 minus 120582120597minus1
119909119906 + 119892 (119905)
= minus119860119906 minus 120582120597minus1
119909119906 minus int
119878
(
1
2
1205882minus 119860119906)119889119909
(81)
We first compute the upper and lower bounds for 119891 forlater use in getting the blow-up result
119891 = minus119860119906 minus 120582int
119909
0
119906 119889119909 minus
1
2
int
119878
1205882119889119909 + 119860int
119878
119906 119889119909
le minus119860119906 minus 120582int
119909
0
119906 119889119909 + 119860int
119878
119906 119889119909 le
119860
2
(1 + 1199062)
+
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le
119860
2
(1 +
1
2
int
119878
(1199062+ 1199062
119909) 119889119909) +
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le 119860 minus
120582
2
+
119860
4
int
119878
(1205882+ 1199062
119909) 119889119909 +
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860
4
1198640+
3119860 minus 2120582
4
[1198901198622119905(int
119878
1199062
0(119909) 119889119909 + 1)]
le 119860 minus
120582
2
+
119860
4
1198640+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(82)
Now we turn to the lower bound of 119891
minus119891 = 119860119906 + 120582int
119909
0
119906 119889119909 +
1
2
int
119878
1205882119889119909 minus 119860int
119878
119906 119889119909
le 119860
1 + 1199062
2
+
1
2
int
119878
1205882119889119909 +
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le 119860 minus
120582
2
+
119860 + 2
4
int
119878
(1205882+ 1199062
119909) 119889119909 +
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860 + 2
4
1198640+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(83)
Combining (82) and (83) we obtain
10038161003816100381610038161198911003816100381610038161003816le 119860 minus
120582
2
+
119860 + 2
4
1198640+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(84)
We know 119872(119905) gt 0 for 119905 isin [0 119879) From the secondequation of (81) we obtain that
120574 (119905) = 120574 (0) 119890minusint119905
0
119872(120591)119889120591 (85)
Hence1003816100381610038161003816120588 (119905 120593 (119905 119909
0))1003816100381610038161003816=1003816100381610038161003816120574 (119905)
1003816100381610038161003816le1003816100381610038161003816120574 (0)
1003816100381610038161003816 (86)
Therefore we have
1198721015840(119905) = 120582119872 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
le 120582119872 (0) +
1
2
1205742(0) +
1
2
1198702
1(119879)
(87)
1198721015840(119905) minus 120582119872 (119905) le
1
2
(sup119909isin119878
1205882
0(119909) + 119870
2
1(119879)) ae 119905 isin [0 119879)
(88)
Integrating (88) on [0 119905] we prove (47) as follows
119872(119905) le sup119909isin119878
1199060119909
(119909) +
1
2
(sup119909isin119878
1205882
0(119909) + 119870
2
1(119879))
119890120582119905minus 1
120582
(89)
To obtain a lower bound for inf119909isin119878
119906119909(119905 119909) we use the
same argumentSince 120590 = 0 (80) becomes
1015840(119905) = 120582 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
1))
1205741015840(119905) = minus120574 ae 119905 isin [0 119879)
(90)
Because of (119905) lt 0 we get from the second equation of(90) that
120574 (119905) = 120574 (0) 119890minusint119905
0
119872(120591)119889120591 (91)
This means that1003816100381610038161003816120588 (119905 120593 (119905 119909
1))1003816100381610038161003816=1003816100381610038161003816120574 (119905)
1003816100381610038161003816ge1003816100381610038161003816120574 (0)
1003816100381610038161003816 (92)
Then
1015840(119905) = 120582 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
ge 120582 (0) +
1
2
1205742(0) +
1
2
1198702
2(119879)
(93)
1015840(119905) minus 120582 (119905) ge
1
2
(inf119909isin119878
1205882
0(119909) minus 119870
2
2(119879)) ae 119905 isin [0 119879)
(94)
Integrating (94) on [0 119905] we prove (48) This completesthe proof of Lemma 14
Lemma 15 Suppose that 120590 isin 119877 0 Suppose 1198830= (1199060
1205880
) isin
119867119904times119867119904minus1 with 119904 ge 2 and let 119879 be the maximal existence time
Journal of Applied Mathematics 9
of the solution 119883 = (119906
120588 ) to (3) with initial data 1198830 Then we
have
120588 (119905 120593 (119905 119909)) 120593119909(119905 119909) = 120588
0(119909) (119905 119909) isin [0 119879] times 119878 (95)
Moreover if there exists119872 gt 0 such that
inf(119905119909)isin[0119879]times119878
120590119906119909(119905 119909) ge minus119872 (119905 119909) isin [0 119879] times 119878 (96)
Then1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin(119878)
=1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin(119878)
le 119890119872119879120590 1003817
1003817100381710038171205880
1003817100381710038171003817119871infin(119878)
(120590 gt 0)
(97)
1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin(119878)
=1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin(119878)
le 119890119873119879120590 1003817
1003817100381710038171205880
1003817100381710038171003817119871infin(119878)
(120590 lt 0)
(98)
where119873 = 1199060119909119871infin(119878)+radic12058221205902minus 1198702
2(119879)120590 minus 120582120590 and119870
2(119879)
is given in (50)
Proof Differentiating the left hand side of (95) with respectto 119905 in view of the relations (12) and (3) we obtain
119889
119889119905
120588 (119905 120593 (119905 119909)) 120593119909(119905 119909)
= [120588119905(119905 120593) + 120588
119909(119905 120593) 120593
119905(119905 119909)]
times 120593119909(119905 119909) + 120588 (119905 120593) 120593
119909119905(119905 119909)
= [120588119905(119905 120593) + 120588
119909(119905 120593) 120593
119905(119905 119909)] 120593
119909(119905 119909)
+ 120588 (119905 120593) 119906119909(119905 120593) 120593
119909(119905 119909)
= [120588119905(119905 120593) + 120588
119909(119905 120593) 120593
119905(119905 119909) + 120588 (119905 120593) 119906
119909(119905 120593)]
times 120593119909(119905 119909) = 0
(99)
This completes the proof of (95) In view of the assump-tion (96) and 120590 gt 0 we obtain 119906(119905 119909) ge minus(119872120590) (119905 119909) isin
[0 119879] times 119878By Lemma 3 and (95) we have
1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin =
1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin =
1003817100381710038171003817100381710038171003817
119890minusint119905
0
119906119909(120591sdot)119889120591
1205880(sdot)
1003817100381710038171003817100381710038171003817119871infin
le 1198901198721198791205901003817
1003817100381710038171205880(sdot)1003817100381710038171003817119871infin
(100)
To obtain (98) we use a similar argument as before Using(13) and the lower bound for 119906
119909(119905 119909) in (46) it follows that
1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin =
1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin =
1003817100381710038171003817100381710038171003817
119890minusint119905
0
119906119909(120591sdot)119889120591
1205880(sdot)
1003817100381710038171003817100381710038171003817119871infin
le 1198901198731198791205901003817
1003817100381710038171205880(sdot)1003817100381710038171003817119871infin
(101)
which proves (98) This completes the proof of Lemma 15
Proof of Theorem 12 Suppose that119879 lt infin and that (42) is notvalid Then there is some positive number119872 gt 0 such that
120590119906119909(119905 119909) ge minus119872 (119905 119909) isin [0 119879] times 119878 (102)
It now follows from Lemma 14 that |119906119909(119905 119909)| le 119862 where
119862 = 119862(119860119872 120590 1198640 120582 119906
0 119879) Therefore Theorem 11 implies
that the maximal existence time 119879 = infin which contradictswith the assumption that 119879 lt infin
Conversely the Sobolev embedding theorem 119867119904(119878) rarr
119871infin(119878) with 119904 gt 12 implies that if (70) holds the corre-
sponding solution blows up in finite time which completesthe proof of Theorem 12
Acknowledgments
The authors would like to thank the referee for commentsand suggestions This work is supported by the NationalNature Science Foundation of China (no 11171135) theNature Science Foundation of the Jiangsu Higher EducationInstitutions of China (no 09KJB110003) and the high-leveltalented person special subsidizes of Jiangsu University (no05JDG047)
References
[1] J K Hunter and R Saxton ldquoDynamics of director fieldsrdquo SIAMJournal on Applied Mathematics vol 51 no 6 pp 1498ndash15211991
[2] Z Yin ldquoOn the structure of solutions to the periodic Hunter-Saxton equationrdquo SIAM Journal on Mathematical Analysis vol36 no 1 pp 272ndash283 2004
[3] H-H Dai and M Pavlov ldquoTransformations for the Camassa-Holm equation its high-frequency limit and the Sinh-Gordonequationrdquo Journal of the Physical Society of Japan vol 67 no 11pp 3655ndash3657 1998
[4] J K Hunter and Y X Zheng ldquoOn a completely integrablenonlinear hyperbolic variational equationrdquo Physica D vol 79no 2ndash4 pp 361ndash386 1994
[5] R Camassa and D D Holm ldquoAn integrable shallow waterequation with peaked solitonsrdquo Physical Review Letters vol 71no 11 pp 1661ndash1664 1993
[6] R S Johnson ldquoCamassa-Holm Korteweg-de Vries and relatedmodels for water wavesrdquo Journal of Fluid Mechanics vol 455pp 63ndash82 2002
[7] A Constantin and B Kolev ldquoOn the geometric approach to themotion of inertialmechanical systemsrdquo Journal of Physics A vol35 no 32 pp R51ndashR79 2002
[8] P J Olver and P Rosenau ldquoTri-Hamiltonian duality betweensolitons and solitary-wave solutions having compact supportrdquoPhysical Review E vol 53 no 2 pp 1900ndash1906 1996
[9] R Beals D H Sattinger and J Szmigielski ldquoInverse scatteringsolutions of the Hunter-Saxton equationrdquo Applicable Analysisvol 78 no 3-4 pp 255ndash269 2001
[10] A Bressan and A Constantin ldquoGlobal solutions of the Hunter-Saxton equationrdquo SIAM Journal on Mathematical Analysis vol94 pp 68ndash92 2010
[11] A Bressan H Holden and X Raynaud ldquoLipschitz metric forthe Hunter-Saxton equationrdquo Journal de Mathematiques Pureset Appliquees vol 94 no 1 pp 68ndash92 2010
10 Journal of Applied Mathematics
[12] J K Hunter and Y X Zheng ldquoOn a nonlinear hyperbolicvariational equation I Global existence of weak solutionsrdquoArchive for Rational Mechanics and Analysis vol 129 no 4 pp305ndash353 1995
[13] J Lenells ldquoThe Hunter-Saxton equation describes the geodesicflow on a sphererdquo Journal of Geometry and Physics vol 57 no10 pp 2049ndash2064 2007
[14] J B Li and Y S Li ldquoBifurcations of travelling wave solutions fora two-component Camassa-Holm equationrdquoActaMathematicaSinica (English Series) vol 24 no 8 pp 1319ndash1330 2008
[15] W Rui and Y Long ldquoIntegral bifurcation method together witha translation-dilation transformation for solving an integrable2-component Camassa-Holm shallow water systemrdquo Journal ofApplied Mathematics vol 2012 Article ID 736765 21 pages2012
[16] H Wu and M Wunsch ldquoGlobal existence for the generalizedtwo-component Hunter-Saxton systemrdquo Journal of Mathemati-cal Fluid Mechanics vol 14 no 3 pp 455ndash469 2012
[17] BMoon andY Liu ldquoWave breaking and global existence for thegeneralized periodic two-component Hunter-Saxton systemrdquoJournal of Differential Equations vol 253 no 1 pp 319ndash3552012
[18] S Wu and Z Yin ldquoBlow-up blow-up rate and decay of thesolution of the weakly dissipative Camassa-Holm equationrdquoJournal ofMathematical Physics vol 47 no 1 Article ID 0135042006
[19] S Wu J Escher and Z Yin ldquoGlobal existence and blow-up phenomena for a weakly dissipative Degasperis-Procesiequationrdquo Discrete and Continuous Dynamical Systems B vol12 no 3 pp 633ndash645 2009
[20] T Kato ldquoQuasi-linear equations of evolution with applicationsto partial differential equationsrdquo in Spectral Theory and Differ-ential Equations vol 448 of Lecture Notes in Mathematics pp25ndash70 Springer Berlin Germany 1975
[21] G Gui and Y Liu ldquoOn the global existence and wave-breakingcriteria for the two-component Camassa-Holm systemrdquo Journalof Functional Analysis vol 258 no 12 pp 4251ndash4278 2010
[22] A Constantin and J Escher ldquoWave breaking for nonlinearnonlocal shallow water equationsrdquo Acta Mathematica vol 181no 2 pp 229ndash243 1998
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Journal of Applied Mathematics
Therefore 1199012(119905) ge 0 for any 119905 isin [0 119879) Since 119909 is chosen
arbitrarily we obtain (46)Let 120590 = 0 Using previous arguments (56) becomes
1198721015840(119905) = minus
120590
2
1198722(119905) + 120582119872 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
1205741015840(119905)
= minus120574119872 ae 119905 isin [0T) (80)
where the notation denotes the derivative with respect to 119905
and 119891 represents the function
119891 = minus119860119906 minus 120582120597minus1
119909119906 + 119892 (119905)
= minus119860119906 minus 120582120597minus1
119909119906 minus int
119878
(
1
2
1205882minus 119860119906)119889119909
(81)
We first compute the upper and lower bounds for 119891 forlater use in getting the blow-up result
119891 = minus119860119906 minus 120582int
119909
0
119906 119889119909 minus
1
2
int
119878
1205882119889119909 + 119860int
119878
119906 119889119909
le minus119860119906 minus 120582int
119909
0
119906 119889119909 + 119860int
119878
119906 119889119909 le
119860
2
(1 + 1199062)
+
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le
119860
2
(1 +
1
2
int
119878
(1199062+ 1199062
119909) 119889119909) +
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le 119860 minus
120582
2
+
119860
4
int
119878
(1205882+ 1199062
119909) 119889119909 +
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860
4
1198640+
3119860 minus 2120582
4
[1198901198622119905(int
119878
1199062
0(119909) 119889119909 + 1)]
le 119860 minus
120582
2
+
119860
4
1198640+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(82)
Now we turn to the lower bound of 119891
minus119891 = 119860119906 + 120582int
119909
0
119906 119889119909 +
1
2
int
119878
1205882119889119909 minus 119860int
119878
119906 119889119909
le 119860
1 + 1199062
2
+
1
2
int
119878
1205882119889119909 +
119860 minus 120582
2
(1 + int
119878
1199062119889119909)
le 119860 minus
120582
2
+
119860 + 2
4
int
119878
(1205882+ 1199062
119909) 119889119909 +
3119860 minus 2120582
4
int
119878
1199062119889119909
le 119860 minus
120582
2
+
119860 + 2
4
1198640+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(83)
Combining (82) and (83) we obtain
10038161003816100381610038161198911003816100381610038161003816le 119860 minus
120582
2
+
119860 + 2
4
1198640+
3119860 minus 2120582
4
[1198901198622119879(10038171003817100381710038171199060
1003817100381710038171003817
2
1198712(119878)+ 1)]
(84)
We know 119872(119905) gt 0 for 119905 isin [0 119879) From the secondequation of (81) we obtain that
120574 (119905) = 120574 (0) 119890minusint119905
0
119872(120591)119889120591 (85)
Hence1003816100381610038161003816120588 (119905 120593 (119905 119909
0))1003816100381610038161003816=1003816100381610038161003816120574 (119905)
1003816100381610038161003816le1003816100381610038161003816120574 (0)
1003816100381610038161003816 (86)
Therefore we have
1198721015840(119905) = 120582119872 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
le 120582119872 (0) +
1
2
1205742(0) +
1
2
1198702
1(119879)
(87)
1198721015840(119905) minus 120582119872 (119905) le
1
2
(sup119909isin119878
1205882
0(119909) + 119870
2
1(119879)) ae 119905 isin [0 119879)
(88)
Integrating (88) on [0 119905] we prove (47) as follows
119872(119905) le sup119909isin119878
1199060119909
(119909) +
1
2
(sup119909isin119878
1205882
0(119909) + 119870
2
1(119879))
119890120582119905minus 1
120582
(89)
To obtain a lower bound for inf119909isin119878
119906119909(119905 119909) we use the
same argumentSince 120590 = 0 (80) becomes
1015840(119905) = 120582 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
1))
1205741015840(119905) = minus120574 ae 119905 isin [0 119879)
(90)
Because of (119905) lt 0 we get from the second equation of(90) that
120574 (119905) = 120574 (0) 119890minusint119905
0
119872(120591)119889120591 (91)
This means that1003816100381610038161003816120588 (119905 120593 (119905 119909
1))1003816100381610038161003816=1003816100381610038161003816120574 (119905)
1003816100381610038161003816ge1003816100381610038161003816120574 (0)
1003816100381610038161003816 (92)
Then
1015840(119905) = 120582 (119905) +
1
2
1205742(119905) + 119891 (119905 120593 (119905 119909
0))
ge 120582 (0) +
1
2
1205742(0) +
1
2
1198702
2(119879)
(93)
1015840(119905) minus 120582 (119905) ge
1
2
(inf119909isin119878
1205882
0(119909) minus 119870
2
2(119879)) ae 119905 isin [0 119879)
(94)
Integrating (94) on [0 119905] we prove (48) This completesthe proof of Lemma 14
Lemma 15 Suppose that 120590 isin 119877 0 Suppose 1198830= (1199060
1205880
) isin
119867119904times119867119904minus1 with 119904 ge 2 and let 119879 be the maximal existence time
Journal of Applied Mathematics 9
of the solution 119883 = (119906
120588 ) to (3) with initial data 1198830 Then we
have
120588 (119905 120593 (119905 119909)) 120593119909(119905 119909) = 120588
0(119909) (119905 119909) isin [0 119879] times 119878 (95)
Moreover if there exists119872 gt 0 such that
inf(119905119909)isin[0119879]times119878
120590119906119909(119905 119909) ge minus119872 (119905 119909) isin [0 119879] times 119878 (96)
Then1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin(119878)
=1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin(119878)
le 119890119872119879120590 1003817
1003817100381710038171205880
1003817100381710038171003817119871infin(119878)
(120590 gt 0)
(97)
1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin(119878)
=1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin(119878)
le 119890119873119879120590 1003817
1003817100381710038171205880
1003817100381710038171003817119871infin(119878)
(120590 lt 0)
(98)
where119873 = 1199060119909119871infin(119878)+radic12058221205902minus 1198702
2(119879)120590 minus 120582120590 and119870
2(119879)
is given in (50)
Proof Differentiating the left hand side of (95) with respectto 119905 in view of the relations (12) and (3) we obtain
119889
119889119905
120588 (119905 120593 (119905 119909)) 120593119909(119905 119909)
= [120588119905(119905 120593) + 120588
119909(119905 120593) 120593
119905(119905 119909)]
times 120593119909(119905 119909) + 120588 (119905 120593) 120593
119909119905(119905 119909)
= [120588119905(119905 120593) + 120588
119909(119905 120593) 120593
119905(119905 119909)] 120593
119909(119905 119909)
+ 120588 (119905 120593) 119906119909(119905 120593) 120593
119909(119905 119909)
= [120588119905(119905 120593) + 120588
119909(119905 120593) 120593
119905(119905 119909) + 120588 (119905 120593) 119906
119909(119905 120593)]
times 120593119909(119905 119909) = 0
(99)
This completes the proof of (95) In view of the assump-tion (96) and 120590 gt 0 we obtain 119906(119905 119909) ge minus(119872120590) (119905 119909) isin
[0 119879] times 119878By Lemma 3 and (95) we have
1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin =
1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin =
1003817100381710038171003817100381710038171003817
119890minusint119905
0
119906119909(120591sdot)119889120591
1205880(sdot)
1003817100381710038171003817100381710038171003817119871infin
le 1198901198721198791205901003817
1003817100381710038171205880(sdot)1003817100381710038171003817119871infin
(100)
To obtain (98) we use a similar argument as before Using(13) and the lower bound for 119906
119909(119905 119909) in (46) it follows that
1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin =
1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin =
1003817100381710038171003817100381710038171003817
119890minusint119905
0
119906119909(120591sdot)119889120591
1205880(sdot)
1003817100381710038171003817100381710038171003817119871infin
le 1198901198731198791205901003817
1003817100381710038171205880(sdot)1003817100381710038171003817119871infin
(101)
which proves (98) This completes the proof of Lemma 15
Proof of Theorem 12 Suppose that119879 lt infin and that (42) is notvalid Then there is some positive number119872 gt 0 such that
120590119906119909(119905 119909) ge minus119872 (119905 119909) isin [0 119879] times 119878 (102)
It now follows from Lemma 14 that |119906119909(119905 119909)| le 119862 where
119862 = 119862(119860119872 120590 1198640 120582 119906
0 119879) Therefore Theorem 11 implies
that the maximal existence time 119879 = infin which contradictswith the assumption that 119879 lt infin
Conversely the Sobolev embedding theorem 119867119904(119878) rarr
119871infin(119878) with 119904 gt 12 implies that if (70) holds the corre-
sponding solution blows up in finite time which completesthe proof of Theorem 12
Acknowledgments
The authors would like to thank the referee for commentsand suggestions This work is supported by the NationalNature Science Foundation of China (no 11171135) theNature Science Foundation of the Jiangsu Higher EducationInstitutions of China (no 09KJB110003) and the high-leveltalented person special subsidizes of Jiangsu University (no05JDG047)
References
[1] J K Hunter and R Saxton ldquoDynamics of director fieldsrdquo SIAMJournal on Applied Mathematics vol 51 no 6 pp 1498ndash15211991
[2] Z Yin ldquoOn the structure of solutions to the periodic Hunter-Saxton equationrdquo SIAM Journal on Mathematical Analysis vol36 no 1 pp 272ndash283 2004
[3] H-H Dai and M Pavlov ldquoTransformations for the Camassa-Holm equation its high-frequency limit and the Sinh-Gordonequationrdquo Journal of the Physical Society of Japan vol 67 no 11pp 3655ndash3657 1998
[4] J K Hunter and Y X Zheng ldquoOn a completely integrablenonlinear hyperbolic variational equationrdquo Physica D vol 79no 2ndash4 pp 361ndash386 1994
[5] R Camassa and D D Holm ldquoAn integrable shallow waterequation with peaked solitonsrdquo Physical Review Letters vol 71no 11 pp 1661ndash1664 1993
[6] R S Johnson ldquoCamassa-Holm Korteweg-de Vries and relatedmodels for water wavesrdquo Journal of Fluid Mechanics vol 455pp 63ndash82 2002
[7] A Constantin and B Kolev ldquoOn the geometric approach to themotion of inertialmechanical systemsrdquo Journal of Physics A vol35 no 32 pp R51ndashR79 2002
[8] P J Olver and P Rosenau ldquoTri-Hamiltonian duality betweensolitons and solitary-wave solutions having compact supportrdquoPhysical Review E vol 53 no 2 pp 1900ndash1906 1996
[9] R Beals D H Sattinger and J Szmigielski ldquoInverse scatteringsolutions of the Hunter-Saxton equationrdquo Applicable Analysisvol 78 no 3-4 pp 255ndash269 2001
[10] A Bressan and A Constantin ldquoGlobal solutions of the Hunter-Saxton equationrdquo SIAM Journal on Mathematical Analysis vol94 pp 68ndash92 2010
[11] A Bressan H Holden and X Raynaud ldquoLipschitz metric forthe Hunter-Saxton equationrdquo Journal de Mathematiques Pureset Appliquees vol 94 no 1 pp 68ndash92 2010
10 Journal of Applied Mathematics
[12] J K Hunter and Y X Zheng ldquoOn a nonlinear hyperbolicvariational equation I Global existence of weak solutionsrdquoArchive for Rational Mechanics and Analysis vol 129 no 4 pp305ndash353 1995
[13] J Lenells ldquoThe Hunter-Saxton equation describes the geodesicflow on a sphererdquo Journal of Geometry and Physics vol 57 no10 pp 2049ndash2064 2007
[14] J B Li and Y S Li ldquoBifurcations of travelling wave solutions fora two-component Camassa-Holm equationrdquoActaMathematicaSinica (English Series) vol 24 no 8 pp 1319ndash1330 2008
[15] W Rui and Y Long ldquoIntegral bifurcation method together witha translation-dilation transformation for solving an integrable2-component Camassa-Holm shallow water systemrdquo Journal ofApplied Mathematics vol 2012 Article ID 736765 21 pages2012
[16] H Wu and M Wunsch ldquoGlobal existence for the generalizedtwo-component Hunter-Saxton systemrdquo Journal of Mathemati-cal Fluid Mechanics vol 14 no 3 pp 455ndash469 2012
[17] BMoon andY Liu ldquoWave breaking and global existence for thegeneralized periodic two-component Hunter-Saxton systemrdquoJournal of Differential Equations vol 253 no 1 pp 319ndash3552012
[18] S Wu and Z Yin ldquoBlow-up blow-up rate and decay of thesolution of the weakly dissipative Camassa-Holm equationrdquoJournal ofMathematical Physics vol 47 no 1 Article ID 0135042006
[19] S Wu J Escher and Z Yin ldquoGlobal existence and blow-up phenomena for a weakly dissipative Degasperis-Procesiequationrdquo Discrete and Continuous Dynamical Systems B vol12 no 3 pp 633ndash645 2009
[20] T Kato ldquoQuasi-linear equations of evolution with applicationsto partial differential equationsrdquo in Spectral Theory and Differ-ential Equations vol 448 of Lecture Notes in Mathematics pp25ndash70 Springer Berlin Germany 1975
[21] G Gui and Y Liu ldquoOn the global existence and wave-breakingcriteria for the two-component Camassa-Holm systemrdquo Journalof Functional Analysis vol 258 no 12 pp 4251ndash4278 2010
[22] A Constantin and J Escher ldquoWave breaking for nonlinearnonlocal shallow water equationsrdquo Acta Mathematica vol 181no 2 pp 229ndash243 1998
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 9
of the solution 119883 = (119906
120588 ) to (3) with initial data 1198830 Then we
have
120588 (119905 120593 (119905 119909)) 120593119909(119905 119909) = 120588
0(119909) (119905 119909) isin [0 119879] times 119878 (95)
Moreover if there exists119872 gt 0 such that
inf(119905119909)isin[0119879]times119878
120590119906119909(119905 119909) ge minus119872 (119905 119909) isin [0 119879] times 119878 (96)
Then1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin(119878)
=1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin(119878)
le 119890119872119879120590 1003817
1003817100381710038171205880
1003817100381710038171003817119871infin(119878)
(120590 gt 0)
(97)
1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin(119878)
=1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin(119878)
le 119890119873119879120590 1003817
1003817100381710038171205880
1003817100381710038171003817119871infin(119878)
(120590 lt 0)
(98)
where119873 = 1199060119909119871infin(119878)+radic12058221205902minus 1198702
2(119879)120590 minus 120582120590 and119870
2(119879)
is given in (50)
Proof Differentiating the left hand side of (95) with respectto 119905 in view of the relations (12) and (3) we obtain
119889
119889119905
120588 (119905 120593 (119905 119909)) 120593119909(119905 119909)
= [120588119905(119905 120593) + 120588
119909(119905 120593) 120593
119905(119905 119909)]
times 120593119909(119905 119909) + 120588 (119905 120593) 120593
119909119905(119905 119909)
= [120588119905(119905 120593) + 120588
119909(119905 120593) 120593
119905(119905 119909)] 120593
119909(119905 119909)
+ 120588 (119905 120593) 119906119909(119905 120593) 120593
119909(119905 119909)
= [120588119905(119905 120593) + 120588
119909(119905 120593) 120593
119905(119905 119909) + 120588 (119905 120593) 119906
119909(119905 120593)]
times 120593119909(119905 119909) = 0
(99)
This completes the proof of (95) In view of the assump-tion (96) and 120590 gt 0 we obtain 119906(119905 119909) ge minus(119872120590) (119905 119909) isin
[0 119879] times 119878By Lemma 3 and (95) we have
1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin =
1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin =
1003817100381710038171003817100381710038171003817
119890minusint119905
0
119906119909(120591sdot)119889120591
1205880(sdot)
1003817100381710038171003817100381710038171003817119871infin
le 1198901198721198791205901003817
1003817100381710038171205880(sdot)1003817100381710038171003817119871infin
(100)
To obtain (98) we use a similar argument as before Using(13) and the lower bound for 119906
119909(119905 119909) in (46) it follows that
1003817100381710038171003817120588 (119905 sdot)
1003817100381710038171003817119871infin =
1003817100381710038171003817120588 (119905 120593 (119905 sdot))
1003817100381710038171003817119871infin =
1003817100381710038171003817100381710038171003817
119890minusint119905
0
119906119909(120591sdot)119889120591
1205880(sdot)
1003817100381710038171003817100381710038171003817119871infin
le 1198901198731198791205901003817
1003817100381710038171205880(sdot)1003817100381710038171003817119871infin
(101)
which proves (98) This completes the proof of Lemma 15
Proof of Theorem 12 Suppose that119879 lt infin and that (42) is notvalid Then there is some positive number119872 gt 0 such that
120590119906119909(119905 119909) ge minus119872 (119905 119909) isin [0 119879] times 119878 (102)
It now follows from Lemma 14 that |119906119909(119905 119909)| le 119862 where
119862 = 119862(119860119872 120590 1198640 120582 119906
0 119879) Therefore Theorem 11 implies
that the maximal existence time 119879 = infin which contradictswith the assumption that 119879 lt infin
Conversely the Sobolev embedding theorem 119867119904(119878) rarr
119871infin(119878) with 119904 gt 12 implies that if (70) holds the corre-
sponding solution blows up in finite time which completesthe proof of Theorem 12
Acknowledgments
The authors would like to thank the referee for commentsand suggestions This work is supported by the NationalNature Science Foundation of China (no 11171135) theNature Science Foundation of the Jiangsu Higher EducationInstitutions of China (no 09KJB110003) and the high-leveltalented person special subsidizes of Jiangsu University (no05JDG047)
References
[1] J K Hunter and R Saxton ldquoDynamics of director fieldsrdquo SIAMJournal on Applied Mathematics vol 51 no 6 pp 1498ndash15211991
[2] Z Yin ldquoOn the structure of solutions to the periodic Hunter-Saxton equationrdquo SIAM Journal on Mathematical Analysis vol36 no 1 pp 272ndash283 2004
[3] H-H Dai and M Pavlov ldquoTransformations for the Camassa-Holm equation its high-frequency limit and the Sinh-Gordonequationrdquo Journal of the Physical Society of Japan vol 67 no 11pp 3655ndash3657 1998
[4] J K Hunter and Y X Zheng ldquoOn a completely integrablenonlinear hyperbolic variational equationrdquo Physica D vol 79no 2ndash4 pp 361ndash386 1994
[5] R Camassa and D D Holm ldquoAn integrable shallow waterequation with peaked solitonsrdquo Physical Review Letters vol 71no 11 pp 1661ndash1664 1993
[6] R S Johnson ldquoCamassa-Holm Korteweg-de Vries and relatedmodels for water wavesrdquo Journal of Fluid Mechanics vol 455pp 63ndash82 2002
[7] A Constantin and B Kolev ldquoOn the geometric approach to themotion of inertialmechanical systemsrdquo Journal of Physics A vol35 no 32 pp R51ndashR79 2002
[8] P J Olver and P Rosenau ldquoTri-Hamiltonian duality betweensolitons and solitary-wave solutions having compact supportrdquoPhysical Review E vol 53 no 2 pp 1900ndash1906 1996
[9] R Beals D H Sattinger and J Szmigielski ldquoInverse scatteringsolutions of the Hunter-Saxton equationrdquo Applicable Analysisvol 78 no 3-4 pp 255ndash269 2001
[10] A Bressan and A Constantin ldquoGlobal solutions of the Hunter-Saxton equationrdquo SIAM Journal on Mathematical Analysis vol94 pp 68ndash92 2010
[11] A Bressan H Holden and X Raynaud ldquoLipschitz metric forthe Hunter-Saxton equationrdquo Journal de Mathematiques Pureset Appliquees vol 94 no 1 pp 68ndash92 2010
10 Journal of Applied Mathematics
[12] J K Hunter and Y X Zheng ldquoOn a nonlinear hyperbolicvariational equation I Global existence of weak solutionsrdquoArchive for Rational Mechanics and Analysis vol 129 no 4 pp305ndash353 1995
[13] J Lenells ldquoThe Hunter-Saxton equation describes the geodesicflow on a sphererdquo Journal of Geometry and Physics vol 57 no10 pp 2049ndash2064 2007
[14] J B Li and Y S Li ldquoBifurcations of travelling wave solutions fora two-component Camassa-Holm equationrdquoActaMathematicaSinica (English Series) vol 24 no 8 pp 1319ndash1330 2008
[15] W Rui and Y Long ldquoIntegral bifurcation method together witha translation-dilation transformation for solving an integrable2-component Camassa-Holm shallow water systemrdquo Journal ofApplied Mathematics vol 2012 Article ID 736765 21 pages2012
[16] H Wu and M Wunsch ldquoGlobal existence for the generalizedtwo-component Hunter-Saxton systemrdquo Journal of Mathemati-cal Fluid Mechanics vol 14 no 3 pp 455ndash469 2012
[17] BMoon andY Liu ldquoWave breaking and global existence for thegeneralized periodic two-component Hunter-Saxton systemrdquoJournal of Differential Equations vol 253 no 1 pp 319ndash3552012
[18] S Wu and Z Yin ldquoBlow-up blow-up rate and decay of thesolution of the weakly dissipative Camassa-Holm equationrdquoJournal ofMathematical Physics vol 47 no 1 Article ID 0135042006
[19] S Wu J Escher and Z Yin ldquoGlobal existence and blow-up phenomena for a weakly dissipative Degasperis-Procesiequationrdquo Discrete and Continuous Dynamical Systems B vol12 no 3 pp 633ndash645 2009
[20] T Kato ldquoQuasi-linear equations of evolution with applicationsto partial differential equationsrdquo in Spectral Theory and Differ-ential Equations vol 448 of Lecture Notes in Mathematics pp25ndash70 Springer Berlin Germany 1975
[21] G Gui and Y Liu ldquoOn the global existence and wave-breakingcriteria for the two-component Camassa-Holm systemrdquo Journalof Functional Analysis vol 258 no 12 pp 4251ndash4278 2010
[22] A Constantin and J Escher ldquoWave breaking for nonlinearnonlocal shallow water equationsrdquo Acta Mathematica vol 181no 2 pp 229ndash243 1998
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Journal of Applied Mathematics
[12] J K Hunter and Y X Zheng ldquoOn a nonlinear hyperbolicvariational equation I Global existence of weak solutionsrdquoArchive for Rational Mechanics and Analysis vol 129 no 4 pp305ndash353 1995
[13] J Lenells ldquoThe Hunter-Saxton equation describes the geodesicflow on a sphererdquo Journal of Geometry and Physics vol 57 no10 pp 2049ndash2064 2007
[14] J B Li and Y S Li ldquoBifurcations of travelling wave solutions fora two-component Camassa-Holm equationrdquoActaMathematicaSinica (English Series) vol 24 no 8 pp 1319ndash1330 2008
[15] W Rui and Y Long ldquoIntegral bifurcation method together witha translation-dilation transformation for solving an integrable2-component Camassa-Holm shallow water systemrdquo Journal ofApplied Mathematics vol 2012 Article ID 736765 21 pages2012
[16] H Wu and M Wunsch ldquoGlobal existence for the generalizedtwo-component Hunter-Saxton systemrdquo Journal of Mathemati-cal Fluid Mechanics vol 14 no 3 pp 455ndash469 2012
[17] BMoon andY Liu ldquoWave breaking and global existence for thegeneralized periodic two-component Hunter-Saxton systemrdquoJournal of Differential Equations vol 253 no 1 pp 319ndash3552012
[18] S Wu and Z Yin ldquoBlow-up blow-up rate and decay of thesolution of the weakly dissipative Camassa-Holm equationrdquoJournal ofMathematical Physics vol 47 no 1 Article ID 0135042006
[19] S Wu J Escher and Z Yin ldquoGlobal existence and blow-up phenomena for a weakly dissipative Degasperis-Procesiequationrdquo Discrete and Continuous Dynamical Systems B vol12 no 3 pp 633ndash645 2009
[20] T Kato ldquoQuasi-linear equations of evolution with applicationsto partial differential equationsrdquo in Spectral Theory and Differ-ential Equations vol 448 of Lecture Notes in Mathematics pp25ndash70 Springer Berlin Germany 1975
[21] G Gui and Y Liu ldquoOn the global existence and wave-breakingcriteria for the two-component Camassa-Holm systemrdquo Journalof Functional Analysis vol 258 no 12 pp 4251ndash4278 2010
[22] A Constantin and J Escher ldquoWave breaking for nonlinearnonlocal shallow water equationsrdquo Acta Mathematica vol 181no 2 pp 229ndash243 1998
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
