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WEAK CONCENTRATION AND WAVE OPERATOR FOR A 3D

COUPLED NONLINEAR SCHRODINGER SYSTEM

ADEMIR PASTOR

Abstract. Reported in this paper are results concerning the Cauchy problem and thedynamics for a cubic nonlinear Schrodinger system arising in nonlinear optics. A sharpcriterium is given concerned with the dichotomy global existence versus finite time blow-up. When a radial solution blows up in finite time, we prove the concentration in thecritical Lebesgue space. Sufficient condition for the scattering and the construction ofthe wave operator in the energy space are also provided.

1. Introduction and Results

This paper is concerned with the following three-dimensional cubic nonlinear Schrodingersystem

iut + ∆u+ (|u|2 + β|v|2)u = 0,

ivt + ∆v + (|v|2 + β|u|2)v = 0,(1.1)

where (x, t) ∈ R3 × R, u = u(x, t) and v = v(x, t) are complex-valued functions, andβ > 0 is a real coupling parameter. System (1.1) appears in many physical situations,especially in nonlinear optics. For instance, when two optical waves of different frequenciescopropagate in a medium and interact nonlinearly through the medium, or when twopolarization components of a wave interact nonlinearly at some central frequency (see [1]).

Note that when v = 0, (1.1) reduces to the well-known nonlinear Schrodinger equation

iut + ∆u+ |u|2u = 0, (1.2)

which is one of the most studied nonlinear dispersive equations. Thus, system (1.1) canalso be thought as a perturbation of the single equation (1.2), in the sense that if thesolution v in (1.1) is small in some sense, then the solution (u, v) will behave as a solutionof (1.2).

Our main goal in this paper is to study the Cauchy problem and the dynamics associatedwith (1.1). So, we accomplish (1.1) with the initial conditions

u(x, 0) = u0(x), v(x, 0) = v0(x). (1.3)

To begin with, note that system (1.1) has some conserved quantities. Indeed, define

M1(u) =

∫|u|2, M2(v) =

∫|v|2, M(u, v) = M1(u) +M2(v), (1.4)

and

E(u, v) =1

2

∫(|∇u|2 + |∇v|2)− 1

4

∫(|u|4 + 2β|uv|2 + |v|4). (1.5)

Then, at least for sufficiently regular solutions, the quantities in (1.4) and (1.5) areconserved by the flow of (1.1). The quantities M and E are frequently referred to asthe mass and the energy associated with (1.1). Note that M and E make sense foru, v ∈ H1 := H1(R3), where H1 denotes the usual L2-based Sobolev space.

2010 Mathematics Subject Classification. Primary 35A01, 35Q53 ; Secondary 35Q35.

1

2 A. PASTOR

System (1.1) is invariant by scaling: assume that (u, v) is a solution of (1.1) with initialdata (u0, v0), then

uλ(x, t) = λu(λx, λ2t), vλ(x, t) = λv(λx, λ2t),

is also a solution with initial data(λu0(λx), λv0(λx)

), for any λ > 0. Now, note that

‖uλ(·, 0)‖L3 + ‖vλ(·, 0)‖L3 = ‖u0‖L3 + ‖v0‖L3

and‖uλ(·, 0)‖H1/2 + ‖vλ(·, 0)‖H1/2 = ‖u0‖H1/2 + ‖v0‖H1/2 ,

where H1/2 is the homogeneous L2-based Sobolev space of order 1/2. Thus, L3 × L3 is

the scale-invariant Lebesgue space and H1/2 × H1/2 is the scale-invariant Sobolev space.Such a spaces are, therefore, called critical spaces.

Let us start with the following local well-posedness result.

Theorem 1.1 (Local well-posedness in the energy space). For any (u0, v0) ∈ H1 ×H1,there are T∗, T

∗ > 0 and a unique solution (u, v) ∈ C((−T∗, T ∗);H1 ×H1) of (1.1) satis-fying (u(0), v(0)) = (u0, v0). Moreover, there holds the blow up alternative:

(i) T ∗ = +∞; or(ii) T ∗ < +∞ and

limt→T ∗

(‖∇u(t)‖22 + ‖∇v(t)‖22) = +∞. (1.6)

A similar statement holds for T∗.

When (i) occurs, we say that the solution is global. When (ii) occurs, we say thatthe solution blows up in finite time T ∗. The proof of Theorem 1.1 is similar to that forthe Schrodinger equation (1.2) and it combines Strichartz estimates with the contractionmapping principle (see [5]).

Once Theorem 1.1 has been established, a natural question present itself: under whatconditions on the initial data (u0, v0) do (i) or (ii) hold? For the Schrodinger equation (1.2)this question has been studied in Refs. [10, 11] (see also Ref. [5] and references therein),where the authors established necessary and sufficient condition for the existence of globalsolutions. These results were obtained taking into account the novel theory put forwardin Ref. [13]. Since then, much effort has been expended on the study of the dynamics ofseveral dispersive equation. In particular, similar questions for nonlinear Schrodinger-typesystems have been addressed (see [4, 6, 16, 17, 18, 22, 24, 25]).

Here, our approach is different from the papers mentioned above. Our analysis isaccomplished directly taking into account the known results for the cubic Schrodingerequation as in [10, 11]. We first prove the following.

Theorem 1.2. Let (u, v) ∈ C((−T∗, T ∗);H1 × H1) be the solution of (1.1) with initialdata (u0, v0) ∈ H1 ×H1, where I := (−T∗, T ∗) is the maximal time interval of existence.Assume that

M(u0, v0)E(u0, v0) < M(P,Q)E(P,Q). (1.7)

The following statements hold.

(i) IfM(u0, v0)(‖∇u0‖22 + ‖∇v0‖22) < M(P,Q)(‖∇P‖22 + ‖∇Q‖22) (1.8)

then

M(u0, v0)(‖∇u(t)‖22 + ‖∇v(t)‖22) < M(P,Q)(‖∇P‖22 + ‖∇Q‖22) (1.9)

and the solution exists globally in time, that is, I = (−∞,∞).

WEAK CONCENTRATION FOR 3D NLS SYSTEM 3

(ii) If

M(u0, v0)(‖∇u0‖22 + ‖∇v0‖22) > M(P,Q)(‖∇P‖22 + ‖∇Q‖22) (1.10)

then

M(u0, v0)(‖∇u(t)‖22 + ‖∇v(t)‖22) > M(P,Q)(‖∇P‖22 + ‖∇Q‖22). (1.11)

Moreover, if u0 and v0 are radial then I is finite and the solution blows up in fintetime.

Here (P,Q) denotes a ground state solution of (1.1) (see Subsection 2.2 below). Theproof of Theorem 1.2 follows the same steps as those in Refs. [11, 13], and the methodcan be applied to several dispersive equations. Partially, Theorem 1.2 has also appearedin Ref. [16]; however, instead of assuming finite variance of the initial data, we assumethat the solution is radial. Hence, in order to prove the blow up in finite time we derive avirial identity similar to that for the Schrodinger equation (see Theorem 2.1). We believethis identity has prospects to be applied in other situations.

Another issue of interest is that of scattering, that is, when the solution of (1.1) approxi-mates a solution of the associated linear system. Roughly speaking, this implies that everynonlinear solution behaves linearly as time evolves. From the mathematical viewpoint thismeans that there is (φ+, ψ+) ∈ H1 ×H1 such that

limt→∞

‖u(t)− U(t)φ+‖H1 + ‖u(t)− U(t)φ+‖H1

= 0. (1.12)

Here and throughout, U(t) denotes the unitary group associated with the linear Schrodingerequation. A criterium to establish the scattering is given below.

Theorem 1.3 (H1 scattering). Assume that (u0, v0) ∈ H1 × H1 satisfies ‖u0‖H1 +‖v0‖H1 ≤ B. Suppose that the IVP (1.1)-(1.3) is globally well-posed in H1 × H1 andK := ‖u‖L5

xt+ ‖v‖L5

xt<∞. Then, there are φ+, ψ+ ∈ H1 such that (1.12) holds.

On the the other hand, one can ask about the following question: given φ+, ψ+ ∈ H1,under what conditions is there (u0, v0) ∈ H1 × H1 such that the corresponding solutiongiven in Theorem 1.1 is global and (1.12) holds? This phenomenon is known as theexistence of wave operators and it is studied in many physical situations (see [26, 27]). Inthis regard, we have the following result.

Theorem 1.4 (Existence of wave operators). Assume (φ+, ψ+) ∈ H1 ×H1 and

1

2(‖φ+‖22 + ‖ψ+‖22)(‖∇φ+‖22 + ‖∇ψ+‖22) < M(P,Q)E(P,Q).

Then, there exist (u0, v0) ∈ H1×H1 such that the solution (u(t), v(t)) of (1.1) with initialcondition (u0, v0) satisfies

limt→∞

‖u(t)− U(t)φ+‖H1 + ‖u(t)− U(t)φ+‖H1

= 0.

In addition,

M(u0, v0)(‖∇u(t)‖22 + ‖∇v(t)‖22) < M(P,Q)(‖∇P‖22 + ‖∇Q‖22),

and

M(u, v) = ‖φ+‖22 + ‖ψ+‖22, E(u, v) =1

2(‖∇φ+‖22 + ‖∇ψ+‖22).

4 A. PASTOR

Finally, we study the weak concentration of the radial finite-time blow up solutions,that is, the concentration of the solution in the critical Lebesgue space L3×L3. The termweak concentration is used to distinguish it from the concentration in L2 × L2. Here,similar to the Schrodinger equation (see [11]) we have two possible scenario for the rate ofconcentration. More precisely, the following holds.

Theorem 1.5. Under the assumptions of Theorem 1.2, if part (ii) occurs then either

(i) there is a constant c > 0, depending on the solution (u, v), such that∫|x|≤c2(‖∇u(t)‖22+‖∇v(t)‖22)−1

(|u(x, t)|3 + |v(x, t)|3

)dx ≥ c−3, (1.13)

or

(ii) there is a sequence tn → T ∗ such that

limn→∞

∫An

(|u(x, tn)|3 + |v(x, tn)|3

)dx =∞, (1.14)

where

An := |x| ≤ c(‖u0‖2 + ‖v0‖2)3/2(‖∇u(tn)‖2 + ‖∇v(tn)‖2)−1/2

and c is a constant independent of (u, v).

Similar considerations apply to T∗.

We do not really have to use the assumptions in Theorem 1.2. As will be clear below,the proof works if one knows that the solution is radial and blows up at some time T ∗. Todemonstrate Theorem 1.5 we follow closely the arguments in Ref. [11, Theorem 1.2].

In Section 2 we introduce some basic tools and give preliminaries results. The rest ofthe paper is devoted to prove Theorems 1.2–1.5.

2. Preliminaries and Basic Tools

First of all, let us introduce some notation used throughout. We use c (or C) to denotevarious constants that may vary line by line. Given any positive numbers A and B, thenotation A . B means that there exists a positive constant c such that A ≤ cB. Given acomplex number z, we use Re(z) and Im(z) to denote, respectively, the real and imaginaryparts of z. By a+ (a−) we indicate a number slightly bigger (smaller) than a. Otherwiseis stated,

∫f always mean integration of the function f over all R3.

We use ‖ · ‖p to denote the Lp(Rn) norm. If the Lp norm is taken over a set Ω ⊂ Rnwe use ‖ · ‖Lp(Ω). Also, if necessary we use subscript to indicate which variable we areintegrating; for instance, ‖f‖Lq

tLrx

stands for the mixed Lebesgue norm ‖‖f(·, t)‖Lrx‖Lq

t,

where the integration in t is over R. When r = q, for short we denote ‖f‖LrtL

rx

by ‖f‖Lrxt

.

If I ⊂ R, in a similar fashion we define ‖f‖LqIL

rx. By Hs = Hs(R3) we designate the usual

L2-based Sobolev space. Also, by Hs = Hs(R3), we represent the homogeneous Sobolev

space. The norms in Hs and Hs will be denote by ‖ · ‖Hs and ‖ · ‖Hs , respectively. TheFourier transform of f is given by

f(ξ) =

∫eixξf(x)dx.


2.1. A Virial Identity. Here we establish a virial identity which will be used to proveTheorem 1.2. Since the result can be proved to a more general system, we consider thefollowing n-dimensional nonlinear Schrodinger system

iut + ∆u+ (|u|2p + β|u|p−1|v|p+1)u = 0,

ivt + ∆v + (|v|2p + β|v|p−1|u|p+1)v = 0,(2.1)

where p ≥ 1 is a real number. Note that system (1.1) is a particular case of (2.1) withp = 1 and n = 3. Our main result here reads as follows.

Theorem 2.1. Let (u, v) ∈ C((−T∗, T ∗);H1(Rn) × H1(Rn)) be a solution of (2.1) andϕ ∈ C∞0 (Rn). Define

V (t) =1

2

∫ϕ(x)(|u|2 + |v|2).

Then,

V ′(t) = Im

∫∇ϕ · ∇uu+ Im

∫∇ϕ · ∇vv (2.2)

and

V ′′(t) =2

n∑k,j=1

Re

∫∂2ϕ

∂xk∂xj(∂xku∂xju+ ∂xkv∂xjv)− 1

2

∫∆2ϕ(|u|2 + |v|2)

+

(1

p+ 1− 1

)∫∆ϕ(|u|2p+2 + 2β|u|p+1|v|p+1 + |v|2p+2),

(2.3)

where ∂xk indicates the partial derivative with respect to xk.

Proof. The proof follows the same steps as in Ref. [12, Lemma 2.9]. So we omit thedetails.

Corollary 2.2. If in addition to the hypothesis of Theorem 2.1, u, v, and ϕ are radiallysymmetric then

V ′′(t) =2

∫ϕ′′(|∇u|2 + |∇v|2

)− 1

2

∫∆2ϕ(|u|2 + |v|2)

+

(1

p+ 1− 1

)∫∆ϕ(|u|2p+2 + 2β|u|p+1|v|p+1 + |v|2p+2)

(2.4)

Proof. It is easy to see, from the fact that u, v, and ϕ are radially symmetric, that

n∑k,j=1

Re∂ϕ

∂xx∂xj(∂xku∂xju+ ∂xkv∂xjv) = ϕ′′

(|∇u|2 + |∇v|2

).

The proof is thus completed.

2.2. Gagliardo-Nirenberg’s inequality and Ground States. In order to obtain thesharp criterium stated in Theorem 1.2 we need a sharp Gagliardo-Nirenberg-type inequal-ity.

Sharp constant for the usual Gagliardo-Nirenberg inequality

‖u‖p+2Lp+2(Rn)

≤ Kp+2best‖∇u‖

np/2L2(Rn)

‖u‖2+p(2−n)/2L2(Rn)

(2.5)

6 A. PASTOR

was first studied in Nagy [21] in the case n = 1 and then for all n ≥ 2 (with 0 < p <4/(n − 2)) in Weinstein [29]. The sharp constant was obtained in terms of the groundstate solution of the semilinear elliptic equation

pn

4∆ψ −

(1 +

p

4(2− n)

)ψ + ψp+1 = 0.

More precisely,

Kp+2best =

p+ 2

2‖ψ‖2L2(Rn)

.

Similar inequalities involving systems have already appeared in the literature. In whatfollows we describe a inequality which generalizes (2.5). These results were established inRef. [8]. Consider the following n-dimensional elliptic system

−∆P + P − (|P |2p + β|P |p−1|Q|p+1)P = 0,

−∆Q+Q− (|Q|2p + β|Q|p−1|P |p+1)Q = 0.(2.6)

Let I : H1(Rn)×H1(Rn)→ R be the functional given by

I(u, v) =1

2(‖∇u‖22+‖u‖22+‖∇v‖22+‖v‖22)− 1

2(p+ 1)(‖u‖2p+2

2p+2+β‖uv‖p+1p+1+‖v‖2p+2

2p+2), (2.7)

where ‖ · ‖q designates the norm in Lq(Rn). It is easy to see that critical points of I areweak solutions of (2.6). Moreover, by elliptic regularity such solutions are indeed smooth.

Recall that a ground state solution of (2.6) is a solution (P,Q) 6= (0, 0) that minimizesI, that is, I(P,Q) ≤ I(u, v) for any solution (u, v) of (2.6). Existence of positive groundstate solutions for (2.6) was investigated in Ref. [19]. In particular such solutions can beobtained by solving the minimization problem

m := infI(u, v) : (u, v) ∈ N, (2.8)

where N ⊂ H1(Rn)×H1(Rn) is the Nehari manifold,

N =

(u, v) 6= (0, 0) : ‖∇u‖22 + ‖u‖22 + ‖∇v‖22 + ‖v‖22 = ‖u‖2p+22p+2 + 2β‖uv‖p+1

p+1 + ‖v‖2p+22p+2

.

Now we are able to state the following Gagliardo-Nirenberg-type inequality.

Proposition 2.3. For all p ≥ 0 and u, v ∈ H1(Rn), it holds

‖u‖2p+22p+2 + 2β‖uv‖p+1

p+1 + ‖v‖2p+22p+2 ≤ Kopt(‖u‖22 + ‖v‖22)p+1−pn/2(‖∇u‖22 + ‖∇v‖22)pn/2, (2.9)

where Kopt is the optimal constant given by

Kopt =2(p+ 1)

(np)np/2(2p+ 2− np)1−np/21

(‖P‖22 + ‖Q‖22)p

=2(p+ 1)pp−np/2

nnp/2(2p+ 2− np)p+1−np/21

mp,

(2.10)

(P,Q) is any ground state solution of (2.6), and m is defined in (2.8).

Proof. The proof was established in Ref. [8, Section 3]. In particular, it was showed thatif (P,Q) is a ground state solution of (2.6), then

m = I(P,Q) =p

2p+ 2− np(‖P‖22 + ‖Q‖22). (2.11)

Thus the second equality in (2.10) holds.


Remark 2.4. Uniqueness of ground states for (2.6) is known only for β sufficiently small[8, 19], in which case either P or Q must vanish identically. However, the second equalityin (2.10) shows that Kopt does not depende on the choice of the ground state.

Remark 2.5. Multiplying the equations in (2.6), respectively, by P and Q, integratingover Rn and using a Pohozaev-type identity, we deduce the following identities (see [8,Section 3])

m =1

n(‖∇P‖22 + ‖∇Q‖22) =

p

2p+ 2(‖P‖2p+2

2p+2 + 2β‖PQ‖p+1p+1 + ‖Q‖2p+2

2p+2). (2.12)

In particular combining (2.11) and (2.12), it is inferred that

‖∇P‖22 + ‖∇Q‖22 =np

2p+ 2− np(‖P‖22 + ‖Q‖22) (2.13)

and

‖P‖2p+22p+2 + 2β‖PQ‖p+1

p+1 + ‖Q‖2p+22p+2 =

2p+ 2

2p+ 2− np(‖P‖22 + ‖Q‖22). (2.14)

In the case of one single equation, let us also recall the following.

Lemma 2.6. The following Gagliardo-Nirenberg inequalities hold.

(i) If u ∈ H1(R3), then

‖f‖44 ≤ C‖∇f‖6(4−p)/(6−p)2 ‖f‖4−6(4−p)/(6−p)

p . (2.15)

(ii) If u ∈ H1(R3) is radially symmetric, then

‖u‖4L4(|x|≥R) ≤C

R2‖u‖3L2(|x|≥R)‖∇u‖L2(|x|≥R). (2.16)

Proof. Part (i) is the usual Gagliardo-Nirenberg inequality [9]. Part (ii) is due to Strauss[28] (see also Ref. [10]).

2.3. Strichartz estimates and Leibniz rule. In order to study the Cauchy problemassociated with (1.1), let us introduce some notation. Given any s ∈ R, we say that a pair

(q, r) is Hs admissible if2

q+

3

r=

3

2− s.

In what follows, we denote

‖u‖S(L2) = sup‖u‖LqtL

rx; (q, r) isL2 admissible, 2 ≤ r ≤ 6, 2 ≤ q ≤ ∞,

and

‖u‖S(H1/2) = sup‖u‖LqtL

rx; (q, r) is H1/2 admissible, 3 ≤ r ≤ 6−, 4+ ≤ q ≤ ∞.

We also need to consider the dual norms:

‖u‖S′(L2) = inf‖u‖Lq′t L

r′x

; (q, r) isL2 admissible, 2 ≤ q ≤ ∞, 2 ≤ r ≤ 6,

and

‖u‖S′(H−1/2) = inf‖u‖Lq′t L

r′x

; (q, r) is H−1/2 admissible,

(4

3

)+

≤ q ≤ 2−, 3+ ≤ r ≤ 6−.

When a time interval I ⊂ R is given, we use S′(L2; I) to inform that the temporalintegral is evaluated over I, that is, we replace ‖ · ‖

Lq′t L

r′x

by ‖ · ‖Lq′I L

r′x

. Similarly to the

other spaces. We now recall the well-known Strichartz inequalities.

8 A. PASTOR

Lemma 2.7. With the above notation we have:

(i) (Linear estimates).‖U(t)u0‖S(L2) . ‖u0‖2

‖U(t)u0‖S(H1/2) . ‖u0‖H1/2 .

(ii) (Inhomogeneous estimates).∥∥∥∥∫ t

0U(t− t′)f(·, t′)dt′

∥∥∥∥S(L2)

. ‖f‖S′(L2),∥∥∥∥∫ t

0U(t− t′)f(·, t′)dt′

∥∥∥∥S(H1/2)

. ‖D1/2f‖S′(L2).

Proof. See Refs. [5, 10].

We also need the Leibniz and chain rules for fractional derivatives.

Lemma 2.8. Let s ∈ (0, 1).

(i) (Fractional chain rule) Suppose G ∈ C1(C). Let 1 < r, r1, r2 < ∞ be such that1/r = 1/r1 + 1/r2. Then,

‖DsG(u)‖r . ‖G′(u)‖r1‖Dsu‖r2 .(ii) (Leibniz’s rule for fractional derivatives) Let 1 < r, p1, p2, q1, q2 < ∞ be such that

1/r = 1/p1 + 1/p2 = 1/q1 + 1/q2. Then,

‖Ds(fg)‖r . ‖f‖p1‖Dsg‖p2 + ‖Dsf‖q1‖g‖q2 .

Proof. See Refs.[7, 14].

Lemma 2.9. Let F (u) = |u|pu with p > 0 and let s ∈ (0, 1). Then, for 1 < q, q1, q2 <∞such that 1/q = 1/q1 + p/q2, we have

‖Ds[F (u+ v)− F (u)]‖q . ‖Dsu‖q1‖v‖q2 + ‖Dsv‖q1‖u+ v‖q2 .

Proof. See Ref.[15].

3. Proof of Theorem 1.2

In this section, we prove Theorem 1.2. We will use the following lemma.

Lemma 3.1. Let I ⊂ R be an open interval containing the origin. Let q > 1, b > 0, anda be real constants. Define γ = (bq)−1/(q−1) and f(r) = a− r+ brq for r ≥ 0. Let G(t) bea continuous nonnegative function on I. Assume that a < (1− 1/q)γ and f G ≥ 0.

(i) If G(0) < γ, then G(t) < γ, for any t ∈ I.(ii) If G(0) > γ, then G(t) > γ, for any t ∈ I.

Proof. This lemma was essentially established in Ref.[2]. We present here the minor mod-ifications in the proof. For (i), since G is continuous and G(0) < γ, there exists ε > 0 suchthat G(t) < γ for t ∈ (−ε, ε). Assume the statement is false. By the continuity of G wethen deduce the existence of t∗ ∈ I with |t∗| ≥ ε such that G(t∗) = γ. Thus,

f G(t∗) = f(γ) = a− γ(1− bγq−1) = a− γ(

1− 1

q

)< 0,

which contradicts de fact that f G ≥ 0. A similar proof holds for (ii). The lemma is thusproved.


Corollary 3.2. Under the assumptions of Lemma 3.1 with a < (1 − 1/q)γ replaced bya < (1− δ1)(1− 1/q)γ, for some small δ1 > 0; if G(0) > γ then there is a δ2 = δ2(δ1) > 0such that G(t) > (1 + δ2)γ.

Proof. Since a−(1−δ1)(1−1/q)γ > a−(1−1/q)γ, we already know from Lemma 3.1 thatG(t) > γ. Now, under the assumption a < (1− δ1)(1− 1/q)γ, there is a ε > 0, dependingon δ1 such that

f(G(t)) ≥ 0 > a− (1− δ1)(1− 1/q)γ = f(γ + ε).

The continuity of G implies that G(t) > γ+ε. By choosing δ2 = ε/γ we obtain the desiredconclusion.

Proof of Theorem 1.2. The idea is to get some control on the gradient of u and v usingthe conserved quantities (1.4) and (1.5) and the Gagliardo-Nirenberg inequality (2.9) withthe sharp constant given in (2.10). Indeed, from (2.9) and (2.10) (with p = 1 and n = 3),

2E(u0, v0) = ‖∇u‖22 + ‖∇v‖22 −1

2(‖u‖44 + 2β‖uv‖22 + ‖v‖44)

≥ ‖∇u‖22 + ‖∇v‖22 −2

33/2

(‖u0‖22 + ‖v0‖22)1/2

‖P‖22 + ‖Q‖22(‖∇u‖22 + ‖∇v‖22)3/2.

(3.1)

Let

a = 2E(u0, v0), b =2

33/2

(‖u0‖22 + ‖v0‖22)1/2

‖P‖22 + ‖Q‖22, and q =

3

2.

Define f(r) = a − r + brq and G(t) = ‖∇u(t)‖22 + ‖∇v(t)‖22. From Theorem 1.1, G iscontinuous. Also, it follows from (3.1) that f G ≥ 0.

In view of (2.13) it is easy to check that G(0) < γ and G(0) > γ are equivalent to (1.8)and (1.9), respectively, where γ is defined as in Lemma 3.1. Moreover, from (2.13) and(2.14), we see that

2E(P,Q) = ‖P‖22 + ‖Q‖22.Hence, a < (1− 1/q)γ is equivalent to (1.7). Lemma 3.1 then gives us the first part of thetheorem.

From now on we assume the radial condition and let us prove that the maximal timeinterval of existence must be finite. As in Refs. [23, 10, 11], we will use the virial identity(2.4). Let ϕ(x) = χ(|x|) be a smooth radial function such that

χ(r) =

r2, 0 ≤ r ≤ 1,

0, r ≥ 3,

and χ′′(r) ≤ 2 for all r ≥ 0. Define χR(r) = R2χ(r/R). By definition, it is easily seenthat if r ≤ R then ∆χR(r) = 6 and ∆2χR(r) = 0.

By choosing ϕ as χR in (2.4), it reduces to

V ′′(t) =2

∫χ′′R(|∇u|2 + |∇v|2

)− 1

2

∫∆2χR(|u|2 + |v|2)

− 1

2

∫∆χR(|u|4 + 2β|u|2|v|2 + |v|4).

(3.2)

We estimate the terms on the right-hand side of (3.2) as follows:

2

∫χ′′R(|∇u|2 + |∇v|2

)≤ 4

∫ (|∇u|2 + |∇v|2

),

10 A. PASTOR

−1

2

∫∆2χR(|u|2 + |v|2) = −1

2

∫|x|≥R

(|u|2 + |v|2) ≤ C

R2

∫|x|≥R

(|u|2 + |v|2),

and

−1

2

∫∆χR(|u|4 + 2β|u|2|v|2 + |v|4)

≤ −3

∫|x|≤R

(|u|4 + 2β|u|2|v|2 + |v|4) + C

∫|x|≥R

(|u|4 + 2β|u|2|v|2 + |v|4)

≤ −3

∫R3

(|u|4 + 2β|u|2|v|2 + |v|4) + C

∫|x|≥R

(|u|4 + 2β|u|2|v|2 + |v|4)

≤ −3

∫R3

(|u|4 + 2β|u|2|v|2 + |v|4) + Cβ

∫|x|≥R

(|u|4 + |v|4),

where we have used Young’s inequality in the last line. Thus, using the conserved quantities(1.4) and (1.5), we can write

V ′′(t) ≤ 12E(u0, v0)− 2(‖∇u‖22 + ‖∇v‖22) +C

R2

∫|x|≥R

(|u|2 + |v|2)

+ Cβ

∫|x|≥R

(|u|4 + |v|4)

≤ 12E(u0, v0)− 2

∫(|∇u|2 + |∇v|2) +

C

R2M(u0, v0) + Cβ

∫|x|≥R

(|u|4 + |v|4).

(3.3)

In view of (2.16) and (1.4), we get

Cβ

∫|x|≥R

(|u|4 + |v|4) ≤CβR2

(‖u0‖32‖∇u‖2 + ‖v0‖32‖∇v‖2)

≤Cβ,εR4‖u0‖62 + ε‖∇u‖22 +

Cβ,εR4‖v0‖62 + ε‖∇v‖22,

(3.4)

where we used the ε-Young’s inequality. The parameter ε will be chosen later. Pluggingthis estimate in (3.3) yields

V ′′(t) ≤ 12E(u0, v0)− (2− ε)(‖∇u‖22 + ‖∇v‖22) +C

R2M(u0, v0)

+Cβ,εR4

(‖u0‖62 + ‖v0‖62).

(3.5)

Multiplying (3.5) by M(u0, v0) we obtain

M(u0, v0)V ′′(t) ≤ 12E(u0, v0)M(u0, v0)− (2− ε)(‖∇u‖22 + ‖∇v‖22)M(u0, v0)

+C

R2M2(u0, v0) +

Cβ,εR4

(‖u0‖62 + ‖v0‖62)M(u0, v0).(3.6)

Since E(u0, v0)M(u0, v0) < E(P,Q)M(P,Q) there is δ1 > 0 such that

E(u0, v0)M(u0, v0) < (1− δ1)E(P,Q)M(P,Q), (3.7)

which in turn is equivalent to the condition a < (1− δ1)γ(1− 1/q) in Corollary 3.2. As aresult, we deduce the existence of a δ2 > 0 such that

M(u0, v0)(‖∇u‖22 + ‖∇v‖22) > (1 + δ2)(‖∇P‖22 + ‖∇Q‖22)M(P,Q), (3.8)


Replacing (3.7) and (3.8) in (3.6), and using that E(P,Q) = (1/6)(‖∇P‖22 + ‖∇Q‖22) (see(2.13)), we have

M(u0, v0)V ′′(t) ≤ [2(1− δ1)− (2− ε)(1 + δ2)](‖∇P‖22 + ‖∇Q‖22)M(P,Q)

+C

R2M2(u0, v0) +

Cβ,εR4

(‖u0‖62 + ‖v0‖62)M(u0, v0).(3.9)

We now can choose ε > 0 small enough and R > 0 large enough such that the right-hand side of (3.9) is bounded by a strictly negative constant. Consequently, the maximalinterval of existence must be finite. This proves the theorem.

When a solution blows up in finite time (not necessarily radial) then a scaling argumentgives a lower bound on the growth of the gradient in (1.6) (a blow-up rate). Indeed,assume that (u, v) blows up in the finite time T ∗. Define,

u(x, s) = λu(λx, t+ λ2s), v(x, s) = λv(λx, t+ λ2s),

where

λ = λ(t) =1

(‖∇u(t)‖2 + ‖∇v(t)‖2)2.

It is easy to see that (u, v) is a solution of (1.1) and

‖∇u(0)‖2 + ‖∇v(0)‖2 = λ1/2(‖∇u(t)‖2 + ‖∇v(t)‖2) = 1.

By observing that Theorem 1.1 also holds here, we deduce that there is s0 > 0 such thatt+ λ(t)2s0 ≤ T ∗. This in turn implies the existence of a constant c > 0 such that

‖∇u(t)‖2 + ‖∇v(t)‖2 ≥c

(T ∗ − t)1/4. (3.10)

As a consequence, we can also prove the blow up in the supercritical Lebesgue spaceLp × Lp, for p > 3. More precisely, we have.

Proposition 3.3. Assume (u0, v0) ∈ H1×H1 and that the corresponding solution (u(t), v(t))blows up in finite time T ∗.

(i) If 3 < p < 4, then

‖u(t)‖p + ‖v(t)‖p ≥C

(T ∗ − t)(p−3)/2p.

(ii) If p ≥ 4, then

‖u(t)‖p + ‖v(t)‖p ≥C

(T ∗ − t)(p−2)/4p.

Proof. The proof is close to that for the Schrodinger equation. The interested reader willfind the details in Ref. [5, Corollary 6.5.14].

4. The Cauchy Problem and proof of Theorem 1.3

We start this section by establishing the global well-posedness for small data in thecritical Sobolev space. The proof of Theorem 1.3 will follow in a similar fashion. As aconsequence, we will also prove that when a solution blows up in finite time T ∗, it alsoblows up in a suitable Strichartz norm (see Corollary 4.2 below).

12 A. PASTOR

Theorem 4.1 (Global well-posedness in the critical space). Assume that (u0, v0) ∈ H1/2×H1/2 satisfies ‖u0‖H1/2 + ‖v0‖H1/2 ≤ A. There is δ > 0 such that if ‖U(t)u0‖S(H1/2) +

‖U(t)v0‖S(H1/2) ≤ δ then the initial-value problem (1.1)-(1.3) is globally well-posed in

H1/2 × H1/2. In addition there existis c > 0 such that

‖u‖S(H1/2) + ‖v‖S(H1/2) ≤ 2(‖U(t)u0‖S(H1/2) + ‖U(t)v0‖S(H1/2)

),

‖D1/2u‖S(L2) + ‖D1/2v‖S(L2) ≤ 2c(‖u0‖H1/2 + ‖v0‖H1/2

).

Proof. The proof is based on the contraction mapping principle. Thus, using the equivalentintegral formulation, the proof reduces to showing that the operator Γ = (Φ,Ψ) given by

Φ(u, v) = U(t)u0 − i∫ t

0U(t− t′)(|u|2u+ β|v|2u)dt′, (4.1)

Ψ(u, v) = U(t)v0 − i∫ t

0U(t− t′)(|v|2v + β|u|2v)dt′, (4.2)

is a contraction on a suitable space. The details are similar to those in the proof ofTheorem 1.3 below (see also Proposition 2.1 in Ref. [10]).

Proof of Theorem 1.3. Since (u, v) is a fixed point of the integral equations (4.1)-(4.2), itfollows that

u(t) = U(t)u0 − i∫ t

0U(t− t′)(|u|2u+ β|v|2u)dt′, (4.3)

v(t) = U(t)v0 − i∫ t

0U(t− t′)(|v|2v + β|u|2v)dt′. (4.4)

Claim. ‖u‖S(L2) + ‖∇u‖S(L2) + ‖v‖S(L2) + ‖∇v‖S(L2) <∞.

Indeed, since ‖u‖L5xt

+ ‖v‖L5xt≤ K, we can decompose [0,∞) =

⋃J0j=1 Ij such that

‖u‖L5IjL5x

+ ‖v‖L5IjL5x< δ, where δ is a small constant to be determined later. It is easy to

see that J0 ∼ K5. In view of Lemma 2.7, we have

‖∇u‖S(L2;Ij) . ‖U(t)∇u0‖S(L2;Ij) + ‖∇(|u|2u)‖L10/7Ij

L10/7x

+ ‖∇(|v|2u)‖L10/7Ij

L10/7x

. ‖u0‖H1 + I1 + I2.(4.5)

Let us estimate I2. We first note that

‖∇(|v|2u)‖L10/7x. ‖v‖2L5

x‖∇u‖

L10/3x

+ ‖∇v‖L10/3x‖v‖L5

x‖u‖L5

x.

Hence, applying Hoder’s inequality, we get

I2 . ‖v‖2L5IjL5x‖∇u‖

L10/3Ij

L10/3x

+ ‖∇v‖L10/3Ij

L10/3x‖v‖L5

IjL5x‖u‖L5

IjL5x. (4.6)

Similarly,

I1 . ‖u‖2L5IjL5x‖∇u‖

L10/3Ij

L10/3x

. (4.7)

From (4.5)-(4.7),

‖∇u‖S(L2;Ij) . ‖u0‖H1 + 2δ2‖∇u‖S(L2;Ij) + δ2‖∇v‖S(L2;Ij). (4.8)

Using the same arguments, we can see that

‖∇v‖S(L2;Ij) . ‖v0‖H1 + 2δ2‖∇v‖S(L2;Ij) + δ2‖∇u‖S(L2;Ij). (4.9)


Summing (4.8) and (4.9),

‖∇u‖S(L2;Ij) + ‖∇v‖S(L2;Ij) . B + 3δ2(‖∇u‖S(L2;Ij) + ‖∇v‖S(L2;Ij)).

By choosing δ small enough, we obtain

‖∇u‖S(L2;Ij) + ‖∇v‖S(L2;Ij) . B. (4.10)

A summation in j gives

‖∇u‖S(L2) + ‖∇v‖S(L2) . BK5.

In view of Holder’s inequality, we also have ‖u‖S(L2) + ‖v‖S(L2) . BK5. The claim is thus

proved.

Now define,

φ+ = u0 − i∫ ∞

0U(−t′)(|u|2u+ β|v|2u)dt′,

ψ+ = v0 − i∫ ∞

0U(−t′)(|v|2v + β|u|2v)dt′.

By using the last claim, it is not difficult to see that φ+, ψ+ ∈ H1.Noting that

u(t)− U(t)φ+ = −i∫ ∞t

U(t− t′)(|u|2u+ β|v|2u)dt′,

v(t)− U(t)ψ+ = −i∫ ∞t

U(t− t′)(|v|2v + β|u|2v)dt′,

and using the previous arguments,

‖u(t)− U(t)φ+‖H1 . ‖u‖2L5[t,∞)

L5x‖u‖

L10/3[t,∞)

L10/3x

+ ‖v‖2L5[t,∞)

L5x‖u‖

L10/3[t,∞)

L10/3x

+ ‖∇u‖L10/3[t,∞)

L10/3x‖u‖2L5

[t,∞)L5x

+ ‖∇u‖L10/3[t,∞)

L10/3x‖v‖2L5

[t,∞)L5x

+ ‖∇v‖L10/3[t,∞)

L10/3x‖v‖L5

[t,∞)L5x‖u‖L5

[t,∞)L5x

(4.11)

The right-hand side of (4.11) goes to zero, as t→∞, in view of the above claim. Similarlywe show that ‖v(t)− U(t)ψ+‖H1 → 0, as t→∞. This proves the theorem.

Corollary 4.2 (Blow up of the Strichartz norm). Assume T ∗ <∞ in Theorem 1.1. Then,

ΛT ∗ := ‖u‖L5[0,T∗]L

5x

+ ‖v‖L5[0,T∗]L

5x

=∞. (4.12)

Proof. Let (u, v) be any solution of (1.1)-(1.3) defined in the interval [0, T ]. We claim that

if ΛT< ∞, then we can extend the solution beyond T . Indeed, by mimicking the proof

of Theorem 1.3, one can see that (u(t), v(t)) ∈ H1 × H1 for all t ∈ [0, T ]. In particular,

(u(T ), v(T )) ∈ H1 × H1. An application of Theorem 1.1 implies the existence of ε0 > 0

such that the solution can be extended to [0, T + ε0). Hence, if T ∗ < ∞, (4.12) musthold.

14 A. PASTOR

5. proof of Theorem 1.4

Following the steps of Theorem 1.3, it suffices to prove that the integral equations

u(t) = U(t)φ+ − i∫ ∞t

U(t− t′)(|u|2u+ β|v|2u)dt′,

v(t) = U(t)ψ+ − i∫ ∞t

U(t− t′)(|v|2v + β|u|2v)dt′,

has a global solution. We first show the existence of a solution for t ≥ T , where T issufficiently large. Note that if δ is sufficiently small we can choose, in view of Theorem4.1, a T > 0 sufficiently large such that

‖U(t)u0‖S(H1/2;[T,∞)) + ‖U(t)v0‖S(H1/2;[T,∞)) < δ.

Thus, as in Theorem 4.1, we can solve the integral equations in H1/2 × H1/2, for t ≥ T .Moreover, the solution (u, v) is small in the sense that

‖u‖S(H1/2;[T,∞)) + ‖v‖S(H1/2;[T,∞)) ≤ 2δ. (5.1)

We now claim that (u(t), v(t)) ∈ H1 × H1. Indeed, taking into account (5.1), as in(4.10), we obtain

‖∇u‖S(L2;[T,∞)) + ‖∇v‖S(L2;[T,∞)) ≤ 2c(‖∇φ+‖2 + ‖∇ψ+‖2).

Dropping the gradient, we deduce a similar estimate. In the same way,

limT→∞

‖∇(u(t)− U(t)φ+)‖S(L2;[T,∞)) = limT→∞

‖∇(v(t)− U(t)ψ+)‖S(L2;[T,∞)) = 0.

This shows that

limt→∞‖∇u(t)‖2 = lim

t→∞‖∇U(t)φ+‖2 = ‖∇φ+‖2, (5.2)

limt→∞‖∇v(t)‖2 = lim

t→∞‖∇U(t)ψ+‖2 = ‖∇ψ+‖2, (5.3)

and

M(u(t), v(t)) = ‖φ+‖22 + ‖ψ+‖22. (5.4)

Next, since ‖u(t)− U(t)φ+‖H1 → 0, as t→∞ and (see Ref. [5, Corollary 2.3.7]),

limt→∞‖U(t)φ+‖4 = 0,

we see from

‖u(t)‖4 ≤ ‖u(t)− U(t)φ+‖4 + ‖U(t)φ+‖4≤ c‖u(t)− U(t)φ+‖H1 + ‖U(t)φ+‖4,

that limt→∞ ‖u(t)‖4 = 0. Similarly, limt→∞ ‖v(t)‖4 = 0. The Young inequality thenimplies

limt→∞

(‖u(t)‖44 + 2β‖u(t)v(t)‖22 + ‖v(t)‖44) = 0. (5.5)

Thus, in view of (5.2)-(5.5),

E(u(t), v(t)) = limt→∞

E(u(t), v(t))

= limt→∞

1

2(‖∇u(t)‖22 + ‖∇v(t)‖22)

=1

2(‖∇φ+‖22 + ‖∇ψ+‖22).


Using the hypothesis we promptly see that, for t ≥ T ,

M(u(t), v(t))E(v(t), u(t)) < M(P,Q)E(P,Q).

In addition, using (2.13),

limt→∞

M(u(t), v(t))(‖∇u(t)‖22 + ‖∇v(t)‖22) = (‖φ+‖22 + ‖ψ+‖22)(‖∇φ+‖22 + ‖∇ψ+‖22)

< 2M(P,Q)E(P,Q)

<1

3M(P,Q)(‖∇P‖22 + ‖∇Q‖22),

where we have used that E(P,Q) = (1/6)(‖∇P‖22 + ‖∇Q‖22). Thus, for T large enough,M(u(T ), v(T ))(‖∇u(T )‖22 +‖∇v(T )‖22) < M(P,Q)(‖∇P‖22 +‖∇Q‖22). From Theorem 1.2,the solution can be extended globally. The proof is thus completed.

6. Proof of Theorem 1.5

In this section we prove Theorem 1.5. Let us start by establishing the nonexistenceof a strong limit at the blow-up time in the critical Lebesgue space even for non-radialsolutions.

Proposition 6.1. Let (u0, v0) ∈ H1 × H1 and assume that the corresponding solutionof the Cauchy problem (1.1)-(1.3) blows up in finite time T ∗. Let tn be a sequence oftimes converging to T ∗, as n→∞. Then, (u(tn), v(tn)) does not have a strong limit inL3 × L3, as n→∞.

Proof. The proof follows the same strategy in Ref.[20], where the authors proved thenonexistence of a strong limit in L2 for the Schrodinger equation. So we leave out thedetails.

Let φ ∈ C∞0 (R3) be a radial cut-off function such that

φ(x) =

1, |x| ≤ 1,

0, |x| ≥ 2.

Let (u, v) be a solution of (1.1). For a fixed R = R(t), to be chosen later, define

u1(x, t) = φ( xR

)u(x, t), u2(x, t) =

(1− φ

( xR

))u(x, t).

Let ψ ∈ C∞0 (R3) be another positive radial cut-off function such that ψ(0) = 1 and ψ = 0for |x| ≥ 1. For a fixed ρ = ρ(t), also to be chosen later, define u1L and u1H via theirFourier transform by

u1L(ξ, t) = ψ

(ξ

ρ

)u1(ξ, t), u1H(ξ, t) =

(1− ψ

(ξ

ρ

))u1(ξ, t).

In a similar manner, we define v1, v2, v1L, and v1H .

Lemma 6.2. Let (u(t), v(t)) ∈ H1×H1 be a solution of (1.1) that blows up in finite timeT ∗. Let

R(t) = c1(‖u0‖2 + ‖v0‖2)3/2(‖∇u(t)‖2 + ‖∇v(t)‖2)−1/2

and

ρ(t) = c2(‖∇u(t)‖22 + ‖∇v(t)‖22),

for suitable chosen constants c1 > 0 and c2 > 0. Under the above notation, we have.

16 A. PASTOR

(i) There is a constant c > 0 such that, as t→ T ∗,

‖u1L‖3 + ‖v1L‖3 ≥ c.

(ii) If ‖u1‖3 + ‖v1‖3 ≤ c∗, then there are c > 0 and x0 = x0(t) such that, as t→ T ∗,

‖u1‖L3(|x−x0(t)|≤ρ−1) + ‖v1‖L3(|x−x0(t)|≤ρ−1) ≥c

(c∗)3.

Moreover, we have |x0(t)| ≤ c(c∗)6ρ(t)−1.

Before proving Lemma 6.2, we establish the following.

Lemma 6.3. Under the above notation,

C(‖u1H‖44 + ‖v1H‖44) ≤ 1

4(β + 1)(‖∇u‖22 + ‖∇v‖22),

where the constant C > 0 appears in the proof of Lemma 6.2 below.

Proof. From the embedding H3/4(R3) → L4(R3) (see for instance Ref.[3, Theorem 6.5.1]),we obtain

C(‖u1H‖44 + ‖v1H‖44) ≤ C‖u1H‖4H3/4 + ‖v1H‖4H3/4

= C

(∫ ∣∣∣|ξ|3/4(1− ψ(ξ/ρ))u1(ξ, t)∣∣∣2 dξ)2

+ C

(∫ ∣∣∣|ξ|3/4(1− ψ(ξ/ρ))v1(ξ, t)∣∣∣2 dξ)2

.

Note, from the mean value theorem, that

|1− ψ(ξ)| = |ψ(0)− ψ(ξ)| ≤ C min1, |ξ|. (6.1)

If |ξ| ≤ ρ, we have

|ξ|3/4|1− ψ(ξ/ρ)| ≤ |ξ|3/4 |ξ|ρ≤ ρ3/4 |ξ|

ρ=|ξ|ρ1/4

. (6.2)

On the other hand, if |ξ| ≥ ρ,

|ξ|3/4|1− ψ(ξ/ρ)| ≤ |ξ|3/4 ≤ ρ3/4 |ξ|1/4

ρ1/4=|ξ|ρ1/4

. (6.3)

Using (6.2) and (6.3) we then see that

C(‖u1H‖44 + ‖v1H‖44) ≤ C

(∫ ∣∣∣∣ |ξ|ρ1/4u1(ξ, t)

∣∣∣∣2 dξ)2

+ C

(∫ ∣∣∣∣ |ξ|ρ1/4v1(ξ, t)

∣∣∣∣2 dξ)2

≤ C 1

ρ(‖|ξ|u1‖42 + ‖|ξ|v1‖42)

≤ C 1

ρ(‖∇u1‖42 + ‖∇v1‖42)

From the definition of u1 and v1, and the fact that ‖∇u(t)‖2 + ‖∇v(t)‖2 →∞, as t→ T ∗

(see Theorem 1.1), it is easy to see that, for t ∼ T ∗, ‖∇u1‖42+‖∇v1‖42 ≤ C(‖∇u‖42+‖∇v‖42)(here and throughout, t ∼ T ∗ means that t is sufficiently close to T ∗). Thus,


C(‖u1H‖44 + ‖v1H‖44) ≤ C 1

ρ(‖∇u‖42 + ‖∇v‖42)

≤ C 1

ρ(‖∇u‖22 + ‖∇v‖22)2

≤ 1

4(β + 1)(‖∇u‖22 + ‖∇v‖22),

where the “suitable” constant c2 in Lemma 6.2 has been chosen large enough so that thelast inequality holds.

We now prove Lemma 6.2.

Proof of Lema 6.2. In view of Theorem 1.1, we know that ‖∇u‖22+‖∇v‖22 →∞, as t→ T ∗.From (1.5), we have

‖u‖44 + 2β‖uv‖22 + ‖v‖44‖∇u‖22 + ‖∇v‖22

= 2− 4E(u0, v0)

‖∇u‖22 + ‖∇v‖22.

Hence, for t ∼ T ∗,‖u‖44 + 2β‖uv‖22 + ‖v‖44 ≥ ‖∇u‖22 + ‖∇v‖22. (6.4)

In view of Young’s inequality and (6.4), we obtain

‖∇u‖22 + ‖∇v‖22 ≤ (β + 1)(‖u‖44 + ‖v‖44)

≤ C(β + 1)(‖u1L‖44 + ‖u1H‖44 + ‖u2‖44 + ‖v1L‖44 + ‖v1H‖44 + ‖v2‖44).(6.5)

An application of Lemma 2.6 (ii) yields

C(‖u2‖44 + ‖v2‖44) ≤ C‖u2‖4L4(|x|≥R) + C‖v2‖4L4(|x|≥R)

≤ C

R2‖u0‖32‖∇u‖2 +

C

R2‖v0‖32‖∇v‖2

≤ C

R2(‖u0‖2 + ‖v0‖2)3(‖∇u‖2 + ‖∇v‖2)

≤ C

c21

(‖∇u‖22 + ‖∇v‖22)

≤ 1

4(β + 1)(‖∇u‖22 + ‖∇v‖22).

(6.6)

Here the constant c1 has also been chosen in such a way that the last inequality holds.A combination of (6.4)-(6.6) with Lemma 6.3 gives

‖∇u‖22 + ‖∇v‖22 ≤ C(β + 1)(‖u1L‖44 + ‖v1L‖44). (6.7)

Next, an application of the Gagliardo-Nirenberg inequality (2.15) leads to

‖∇u‖22 + ‖∇v‖22 ≤ C(β + 1)(‖u1L‖23‖∇u1L‖22 + ‖v1L‖23‖∇v1L‖22)

≤ C(β + 1)(‖u1L‖3 + ‖v1L‖3)2(‖∇u‖22 + ‖∇v‖22).(6.8)

Therefore, inequality (6.8) promptly imply part (i) of the lemma.

18 A. PASTOR

To prove part (ii), we note that from (6.7),

‖∇u‖22 + ‖∇v‖22 ≤ c(‖u1L‖∞‖u1L‖33 + ‖v1L‖∞‖v1L‖33)

≤ c(c∗)3 supx∈R3

(|(ψ(·/ρ))∨ ∗ u1(x)|+ |(ψ(·/ρ))∨ ∗ v1(x)|

).

(6.9)

By choosing x0 = x0(t) such that 1/2 of this supremum is attained, and using Holder’sinequality we arrive to

‖∇u‖22 + ‖∇v‖22 ≤ c(c∗)3∣∣∣ρ3

∫ψ(ρ(x0 − y))

(u1(y, t) + v1(y, t)

)dy∣∣∣

≤ c(c∗)3ρ3

∫ρ|x0−y|≤1

(|u1(y, t)|+ |v1(y, t)|

)dy

≤ c(c∗)3ρ(‖u1‖L3(|x0−x|≤ρ−1) + ‖v1‖L3(|x0−x|≤ρ−1)

).

(6.10)

The definition of ρ yields the inequality claimed in part (ii). To see that |x0(t)| ≤c(c∗)6ρ(t)−1 one proceeds as in Ref.[11, Proposition 3.1].

Finally, we prove Theorem 1.5.

Proof of Theorem 1.5. Since ‖u1L‖3 + ‖v1L‖3 ≤ ‖u1‖3 + ‖v1‖3, it follows from Lemma 6.2that ‖u1(t)‖3 + ‖u1(t)‖3 is bounded from below. We now divide the proof into two cases.

Case 1. If ‖u1(t)‖3 + ‖v1(t)‖3 is not bounded.In this case, there is a sequence tn → T ∗ such that ‖u1(tn)‖3 + ‖v1(tn)‖3 →∞. Since

‖u1(tn)‖33 + ‖v1(tn)‖33 =

∫|x|≤2R(tn)

∣∣∣φ( x

R(tn)

)u(x, tn)

∣∣∣3 +∣∣∣φ( x

R(tn)

)v(x, tn)

∣∣∣3dx≤∫|x|≤2R(tn)

∣∣∣u(x, tn)∣∣∣3 +

∣∣∣v(x, tn)∣∣∣3dx,

(6.11)

taking the limit in (6.11), as n→∞, we obtain (1.14).

Case 2. If there is c∗ > 0 such ‖u1(t)‖3 + ‖u1(t)‖3 ≤ c∗.Here, Lemma 6.2 part (ii) implies that

c

(c∗)3≤

(∫|x−x0|≤ρ−1

|u1(x, t)|3dx

)1/3

+

(∫|x−x0|≤ρ−1

|v1(x, t)|3dx

)1/3

≤

(∫|x|≤|x0|+ρ−1

|u1(x, t)|3dx

)1/3

+

(∫|x|≤|x0|+ρ−1

|v1(x, t)|3dx

)1/3(6.12)

Since for t ∼ T ∗, |x0|+ ρ−1 ≤ c(c∗)6ρ−1 + ρ−1 = c(c∗)6ρ−1. Inequality (6.12) gives (1.13)and the proof of the theorem is completed.

Remark 6.4. Note that Case 1 in the proof of Theorem 1.5 includes the case when either‖u1(t)‖3 or ‖v1(t)‖3 is unbounded. It should be interesting to know when only one of thesequantities is unbounded.

Acknowledgements

The author is partially supported by CNPq-Brazil and FAPESP-Brazil.


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20 A. PASTOR

IMECC-UNICAMP, Rua Sergio Buarque de Holanda, 651, 13083-859, Campinas-SP, BrazilE-mail address: [email protected]

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