Electronic Journal of Qualitative Theory of Differential Equations2019, No. 68, 1–14; https://doi.org/10.14232/ejqtde.2019.1.68 www.math.u-szeged.hu/ejqtde/

1

Weak upper semicontinuity of pullback attractors fornonautonomous reaction-diffusion equations

Jacson Simsen

Instituto de Matemática e Computação, Universidade Federal de Itajubá,Itajubá, 37500-903, Minas Gerais, Brazil

Received 7 January 2019, appeared 18 September 2019

Communicated by Christian Pötzsche

Abstract. We consider nonautonomous reaction-diffusion equations with variable ex-ponents and large diffusion and we prove continuity of the flow and weak upper semi-continuity of a family of pullback attractors when the exponents go to 2 in L∞(Ω).

Keywords: ODE limit problems, nonautonomous reaction-diffusion equations,parabolic problems, variable exponents, pullback attractors, upper semicontinuity.

2010 Mathematics Subject Classification: 35B40, 35B41, 35K57, 35K59.

1 Introduction

In this work, we will study the following problem∂us∂t (t)−div(Ds|∇us(t)|ps(x)−2∇us(t)) + C(t)|us(t)|ps(x)−2us(t) = B(t, us(t)), t > τ,

us(τ) = uτs,(1.1)

under homogeneous Neumann boundary conditions, uτs ∈ H := L2(Ω), Ω ⊂ Rn (n ≥ 1) is asmooth bounded domain, Ds ∈ [1, ∞), ps(·) ∈ C(Ω), p−s := minx∈Ω ps(x) > 2, and there existsa constant a > 2 such that p+s := maxx∈Ω ps(x) ≤ a, for all s ∈ N. We assume that Ds → ∞and ps(·)→ 2 in L∞(Ω) as s→ ∞. The terms B and C are assumed to satisfy:

Assumption B The mapping B : R× H → H is such that

(B1) there exists L ≥ 0 such that

‖B(t, y1)− B(t, y2)‖H ≤ L‖y1 − y2‖H,

for all t ∈ R and y1, y2 ∈ H.

(B2) for all y ∈ H the mapping t 7→ B(t, y) belongs to L2(τ, T; H).

1Email: jacson@unifei.edu.br

2 J. Simsen

(B3) the function t 7→ ‖B(t, 0)‖H is nondecreasing, absolutely continuous and bounded oncompact subsets of R.

Assumption C C(·) ∈ L∞([τ, T]; R+) is monotonically nonincreasing in time and it is boundedfrom above and below, let us consider 0 < α ≤ C(t) ≤ M, ∀ t ∈ R, for some positive constantsα and M. The constants α and M are taken uniform on τ and T.

The aim of this work is to study the asymptotic behavior of the solutions as s → ∞.We prove continuity of the flow and weak upper semicontinuity of the family of pullbackattractors as s goes to infinity for the problem (1.1) with respect to the couple of parameters(Ds, ps), where ps is the variable exponent and Ds is the diffusion coefficient.

It is a well-known fact that reaction-diffusion systems are used for many models of chem-ical, biological and ecological problems. When variable exponents are included these modelsoften appear in applications in electrorheological fluids [8, 9, 20–22] and image processing[6, 11].

Reaction-diffusion systems for which the flow is essentially determined by an ordinarydifferential equation have been studied by many researchers and they often appear as shadowsystems. Large diffusion phenomena have application in chemical fluid flows, see for example[19]. Recently an application was given to describing algal blooms [17]. Semilinear reaction-diffusion equations for large diffusion have been considered in many works, see for examplethe following works and the references therein [1, 3, 4, 7, 12–14, 32]. Moreover, quasilinearreaction-diffusion equations with large diffusion have been considered in many works forp-Laplacian problems, see for example [2, 24, 25, 28] and the references therein.

The study of the continuity with respect to initial conditions and parameters is importantto verify the stability of a PDE model. In [23,26–28] the authors investigated in which way theexponent parameter p(x) affects the dynamic of PDEs involving the p(x)-Laplacian. In [23,26, 27] the limit problem was also a PDE and in [28] the limit problem was an ODE.

In [10] the authors considered the following nonautonomous equation∂us∂t (t)− div(Ds|∇us|ps(x)−2∇us) + C(t)|us|ps(x)−2us = B(us(t)), t > τ,

us(τ) = uτs,(1.2)

under homogeneous Neumann boundary conditions, uτs ∈ H := L2(Ω), Ω ⊂ Rn (n ≥ 1) is asmooth bounded domain, B : H → H is a globally Lipschitz map with Lipschitz constant L ≥0, Ds ∈ [1, ∞), C(·) ∈ L∞([τ, T]; R+) is bounded from above and below and is monotonicallynonincreasing in time, ps(·) ∈ C(Ω), p−s := minx∈Ω ps(x) ≥ p, p+s := maxx∈Ω ps(x) ≤ a,for all s ∈ N, when ps(·) → p in L∞(Ω) and Ds → ∞ as s → ∞, with a, p > 2 positiveconstants. They proved continuity of the flows and upper semicontinuity of the family ofpullback attractors.

In this paper we will give one step more and reach the linear case, i.e., ps(·)→ 2 in L∞(Ω)

as s → ∞. A revision of the paper [10] shows that, with the assumptions given on B, thedifference of the explicit dependence on time on the reaction term B(t, us(t)) is unimportant inorder to obtain all the results included in that work, in particular, the existence of solution andpullback attractors. It is worth to mention that external forcing terms satisfying Assumption Bwere already considered in the works [16,30]. Problem (1.1) has a strong solution us, i.e., us ∈C([τ, T]; H) is absolutely continuous in any compact subinterval of (τ, T), us(t) ∈ D(As(t))for a.e. t ∈ (τ, T), and

dus

dt(t) + As(t)(us(t)) = B(t, us(t)) for a.e. t ∈ (τ, T),

Weak upper semicontinuity of pullback attractors 3

where As(t)(us) := −div(Ds|∇us|ps(x)−2∇us) + C(t)|us|ps(x)−2us and problem (1.1) has a pull-back attractor Us = As(t)t∈R (see [10]). We will use the technique developed in [29] forautonomous problems and make the mutatis mutandis over it to deal with the nonautonomousproblem in order to prove a weak upper semicontinuity of the family of pullback attractorsUss∈N as s goes to infinity for the problem (1.1), weak in the sense that we control thegaps between two consecutive exponent functions for a given δ0 (see condition (H2) in Sec-tion 3) in order to obtain, for each ` ∈ R, As(`) ⊂ Oδ0(A∞(`)), for s large enough, whereU∞ = A∞(t)t∈R will be the pullback attractor of the limit problem.

Considering ps(·) → 2 in L∞(Ω) and large diffusion, a fast redistribution process of thesolution occurs having homogenization, any spatial variation of the solution is reduced tozero; i.e.; the only relevant parameter at the limit of the dynamics of the problem becomes thetime. In other words, the limit problem will be the nonautonomous ODE (2.2). For this reasonwe will consider a family (in p) of ODE’s reaching the same limit problem (2.2) when p goesto 2. So, we will consider the following hypothesis

There exists ε0 > 0 such that if ps ∈ Fε0(2) := g; ‖g− 2‖L∞(Ω) ≤ ε0,then ps is a constant function.

(H)

The paper is organized as follows. In Section 2 we prove a uniform estimate for thesolutions of nonlinear ODEs and we prove continuity of the solutions with respect to initialconditions and exponent parameters. In Section 3 we prove that the solutions us of the PDE(1.1) converge for s→ ∞ to the solution u of the limit problem (2.2) which is an ODE, and, afterthat, we obtain a weak upper semicontinuity of the pullback attractors for the problem (1.1).

2 The family of nonautonomous ODEs and its limit problem

Now consider the following family (in p) of ODEsup(t) + C(t)|up(t)|p−2up(t) = f (t, up(t)), t > τ,

up(τ) = uτp ∈ R,(2.1)

with p ∈ (2, 3] a constant and f : R×R→ R satisfying

(i) | f (t, x1)− f (t, x2)| ≤ L|x1 − x2|, for all t ∈ R and x1, x2 ∈ R.

(ii) for all x ∈ R the mapping t 7→ f (t, x) belongs to L2(τ, T; R).

(iii) the function t 7→ | f (t, 0)| is nondecreasing, absolutely continuous and bounded on com-pact subsets of R.

With the assumptions given on f , the explicit dependence on time on the reaction termis unimportant in order to obtain existence of solution and pullback attractors. With thesame arguments as in Sections 5 and 6 in [10] problem (2.1) has a unique strong solutionup and has a pullback attractor Vp = Mp(t)t∈R. Nonautonomous ODEs had appeared inchemotherapy models, see [15].

Now, we intend to study the sensitivity of problem (2.1) when the constant exponent pgoes to 2. We guess and will prove that the limit problem is

u(t) + C(t)u(t) = f (t, u(t)), t > τ,

u(τ) = uτ ∈ R.(2.2)

4 J. Simsen

It is straightforward to check the abstract conditions in [18] for our problem (2.2) in orderto obtain the existence of a classical unique global solution u for (2.2). Moreover, given T > τ

and uτ ∈ R, there exists a constant K∞ = K∞(uτ, T) > 0 such that |u(t)| ≤ K∞, for all t ∈ [τ, T].In the next result we prove the continuity of the solutions of (2.1) with respect to the initial

data and exponent parameter.

Theorem 2.1. Let up be a solution of (2.1) with up(τ) = uτp and let u be the solution of (2.2) withu(τ) = uτ. If uτp → uτ in R as p→ 2, then for each T > τ, up → u in C([τ, T]; R) as p→ 2.

Proof. Let T > τ be fixed and suppose that uτp → uτ in R as p → 2. Subtracting the twoequations in (2.1) and (2.2) and making the product with up − u we obtain

12

ddt|up(t)− u(t)|2 + C(t)[|up(t)|p−2up(t)− u(t)][up(t)− u(t)]

= [ f (t, up(t))− f (t, u(t))][up(t)− u(t)].

Adding ±C(t)|u(t)|p−2u(t), using that f is Lipschitz with respect to the second variable andthat for any ξ, η ∈ Rn,

(|ξ|p−2ξ − |η|p−2η)(ξ − η) ≥ 0, M ≥ C(t) ≥ α ∀ t ∈ [τ, T]

we obtain

12

ddt|up(t)− u(t)|2 ≤ L|up(t)− u(t)|2 − C(t)(|u(t)|p−2 − 1)u(t)(up(t)− u(t))

≤ L|up(t)− u(t)|2 + M∣∣∣|u(t)|p−1 − |u(t)|

∣∣∣ |up(t)− u(t)|,

for all t ∈ (0, T).Now, let us estimate the term∣∣∣|u(t)|p−1 − |u(t)|

∣∣∣ |up(t)− u(t)|.

By the Mean Value Theorem, for each p > 2 there is a q ∈ (2, p) such that∣∣∣|u(t)|p−1 − |u(t)|∣∣∣ = ∣∣∣|u(t)|q−1 ln |u(t)|

∣∣∣ |p− 2|

provided that u(t) 6= 0. Consider the continuous function gθ : [0, K∞]→ R given by

gθ(w) =

wθ ln w if w ∈ (0, K∞]

0 if w = 0,

where θ ≥ 1 is a given number. Using this continuous function defined in the compact set[0, K∞] with θ = 1 when |u(t)| < 1 and with θ = 2 when |u(t)| ≥ 1, there exists a positiveconstant R such that ∣∣∣|u(t)|q−1 ln |u(t)|

∣∣∣ ≤ R,

for all t ∈ [τ, T] with u(t) 6= 0. So,∣∣∣|u(t)|p−1 − |u(t)|∣∣∣ ≤ R|p− 2|,

Weak upper semicontinuity of pullback attractors 5

for all t ∈ [τ, T]. Thus,

12

ddt|up(t)− u(t)|2 ≤ L|up(t)− u(t)|2 + MR|p− 2||up(t)− u(t)|

≤ L|up(t)− u(t)|2 + 12[MR|p− 2|]2 + 1

2|up(t)− u(t)|2,

for all t ∈ (τ, T).Integrating from τ to t, t ≤ T, we obtain

|up(t)− u(t)|2 ≤|uτp − uτ|2 + [MR|p− 2|]2(T − τ) +∫ t

τ(2L + 1)|up(τ)− u(τ)|2dτ.

So, by Gronwall-Bellman’s Lemma we obtain

|up(t)− u(t)|2 ≤[|uτp − uτ|2 + (MR|p− 2|)2(T − τ)

]e(2L+1)(T−τ),

for all t ∈ [τ, T]. Therefore, up → u in C([τ, T]; R) as p→ 2.

If we restrict the initial conditions to a bounded setM⊂ R in problem (2.1) and considerL < α then we can obtain the following uniform estimates of the solutions of problem (2.1).

Proposition 2.2. Consider f with Lipschitz constant L < α, where α is from Assumption C. LetMbe a bounded set and up be a solution of (2.1) with up(τ) = uτp ∈ M. There exists a positive numberr0 such that |up(t)| ≤ r0, for each t ≥ τ and for all p ∈ (2, 3].

Proof. Let t > τ. Multiplying the equation on (2.1) by up(t) we have that

12

ddτ|up(t)|2 ≤ −C(t)|up(t)|p + | f (t, up(t))||up(t)|

≤ −α|up(t)|p + | f (t, up(t))− f (t, 0)||up(t)|+ | f (t, 0)||up(t)|.

So,12

ddτ|up(t)|2 ≤ −α|up(t)|p + L|up(t)|2 + C0|up(t)|, (2.3)

where C0 := supt∈[τ,T] | f (t, 0)| ≥ 0.If |up(t)| > 1, −|up(t)|p ≤ −|up(t)|2, then from (2.3)

12

ddτ|up(t)|2 ≤ (L− α)|up(t)|2 + C0|up(t)|.

Consider ε > 0 arbitrary. Using Young’s inequality we obtain

12

ddτ|up(t)|2 ≤

(−α + L +

12

ε2)|up(t)|2 +

12

(C0

ε

)2

.

Now, choosing ε = ε1 > 0 sufficiently small such that 0 < ε1 < (α− L)1/2 we obtain

12

ddτ|up(t)|2 ≤ −β|up(t)|2 + C1,

where β := α2 −

L2 > 0 and C1 := 1

2

(C0ε1

)2. Then

ddτ

[|up(t)|2]e2βt + 2β|up(t)|2e2βt ≤ 2C1e2βt. (2.4)

6 J. Simsen

If |up(t)| ≤ 1, then from (2.3),

ddτ|up(t)|2 ≤ 2(L + C0) =: C2.

Thus,d

dτ[|up(t)|2]e2βτ + 2β|up(t)|2e2βt ≤ C2e2βt + 2β|up(t)|2e2βt ≤ (C2 + 2β)e2βt. (2.5)

Considering yp(t) := |up(t)|2 and C3 := max2C1, C2 + 2β, we obtain from (2.4) and (2.5)that

ddτ

[yp(t)e2βt] ≤ C3e2βt, for all t > τ.

Integrating from τ to `, we have

yp(`)e2β` ≤ yp(τ)e2βτ +C3

2βe2β` − C3

2βe2βτ ≤ |uτp|2e2βτ +

C3

2βe2β`.

Multiplying by e−2β`, we obtain

|up(`)|2 = yp(`) ≤ |uτp|2e−2β(`−τ) +C3

2βe0 ≤ |uτp|2e0 +

C3

2β, for all ` ≥ τ.

Since uτp ∈ M and M is bounded, there exists K ≥ 0 such that |uτp| ≤ K for all p ∈ (2, 3].Thus,

|up(`)| ≤ r0 :=(

K2 +C3

2β

)1/2

,

for all ` ≥ τ and p ∈ (2, 3].

3 Continuity of the flow and weak upper semicontinuity of attrac-tors

Our objective in this section is to prove that the limit problem of problem (1.1) as Ds increasesto infinity and ps(·)→ 2 in L∞(Ω) as s→ ∞ is described by the ordinary differential equationin (2.2).

The next result guarantees that (2.2) is in fact the limit problem for (1.1), as s → ∞. Theproof is analogous to the proof of Theorem 5.3 in [10].

Theorem 3.1. Let us be a solution of (1.1) with us(τ) = uτs and let u be the solution of (2.2) withf = B|R×R and u(τ) = uτ. If uτs → uτ in H as s→ ∞, then for each T > τ, us → u in C([τ, T]; H)

as s→ +∞.

Let us now review some concepts and results on processes.

Definition 3.2. An evolution process in a metric space X is a family S(t, τ) : X → Xt≥τ ofcontinuous maps satisfying:

(i) S(τ, τ) = I (here I denotes the identity operator);

(ii) S(t, τ) = S(t, s)S(s, τ), τ ≤ s ≤ t.

Weak upper semicontinuity of pullback attractors 7

Definition 3.3. Let S(t, τ)t≥τ be an evolution process in a metric space X. Given A and Bsubsets of X, we say that A pullback attracts B at time t if

limτ→−∞

dist (S(t, τ)B, A) = 0,

where dist denote the Hausdorff semi-distance.

Definition 3.4. A family of subsets A(t) : t ∈ R of X is invariant relatively to the evolutionprocess S(t, τ)t≥τ if S(t, τ)A(τ) = A(t) for any t ≥ τ.

Definition 3.5. A family of subsets A(t) : t ∈ R of X is a pullback attractor for the evolutionprocess S(t, τ)t≥τ if it is invariant, A(t) is compact for each t ∈ R, pullback attracts allbounded subsets of X at time t for each t ∈ R and it is the minimal among all closed familieswhich pullback attracts bounded sets of X.

Definition 3.6. A process S(·, ·) in a metric space X is said to be pullback asymptoticallycompact if, for each t ∈ R, each sequence sk ≤ t with sk → −∞ as k → ∞, and eachbounded sequence xk in X, the sequence S(t, sk)xk has a convergent subsequence.

Definition 3.7. We say that a process S(·, ·) is pullback bounded dissipative if there exists afamily B(·) of bounded sets such that B(t) pullback attracts bounded sets at time t, for eacht ∈ R.

Definition 3.8. We say that a process S(·, ·) is strongly pullback bounded dissipative if for eacht ∈ R there is a bounded subset B(t) of X that pullback attracts bounded subsets of X at time τ

for each τ ≤ t; i.e., given a bounded subset D of X and τ ≤ t, lims→−∞ dist(S(τ, s)D, B(t)) = 0.

Theorem 3.9 ([5, Theorem 2.23]). If a process S(·, ·) is strongly pullback bounded dissipative andpullback asymptotically compact, then S(·, ·) has a compact pullback attractor.

Theorem 3.10. Consider f with Lipschitz constant L < α, where α is from Assumption C. Theproblem (2.2) defines a pullback asymptotically compact process.

Proof. Let t > τ. We define S(t, τ) : R→ R by S(t, τ)uτ = u(t) with u being the unique globalsolution of the problem (2.2) with u(τ) = uτ. It is easy to see that S(t, τ) : R → R, t ≥ τverifies the process properties. Multiplying the equation in (2.2) by u(t) and proceeding ina completely analogous way as in the proof of Proposition 2.2 we obtain that there exists apositive number r0 such that |u(t)| ≤ r0, for each t ≥ τ(with the same constant r0). Thus,we conclude that for each t > τ, S(t, τ) maps bounded sets into bounded sets and the resultfollows.

Observe that the process defined by the problem (2.2) is not necessarily pullback boundeddissipative. If we consider the very simple example, C(t) ≡ 1 and f : R×R → R given byf (t, u) := ηu with η > 1 a real number and so the solution of (2.2) is u(t) = uτe(η−1)(t−τ)

and |u(t)| → ∞ as τ → −∞. In this case a pullback attractor for the problem (2.2) doesnot exist. There are examples that provide situations where the process defined by the limitproblem (2.2) is pullback bounded dissipative. If C(t) ≡ 1 and f : R × R → R given byf (t, u) := ηu with η < 1 a real number then the solution of (2.2) is u(t) = uτe(η−1)(t−τ) andu(t) → 0 as τ → −∞. So, the process defined by the limit problem (2.2) is strongly pullbackbounded dissipative.

Now, we suppose that f = B|R×R : R×R → R, is such that the limit problem (2.2) hasa strongly pullback bounded dissipative process. So, let U∞ = A∞(t)t∈R be the pullbackattractor for (2.2) with f = B|R×R.

We need to use the following

8 J. Simsen

Theorem 3.11 ([10]). Let Us = As(t)t∈R be the pullback attractor associated with problem (1.1)and Vp = Mp(t)t∈R the pullback attractor for problem (2.1) with f = B|R×R. Then, for each` ∈ R, we have dist(As(`);Mp(`))→ 0 in the topology of H, when ps(·)→ p > 2 in L∞(Ω).

Let us consider ` ∈ R arbitrarily fixed. The condition (H) is needed in the proof of theweak upper semicontinuity of the family of pullback attractors for problem (1.1) as ps → 2in L∞(Ω). Moreover, after the functions ps(·) enter into Fε0(2), given δ0 > 0, in order toshow As(τ) ⊂ Oδ0(A∞(τ)) for s > 0 large enough, we have to control the gap between twoconsecutive functions ps and ps+1 by an appropriate term which depends on s and δ0 (seehypothesis (H2) below).

Consider p := 2 + ε0, where ε0 > 0 is from hypothesis (H). Then there exists s1 ∈N largeenough such that 2 < ps1 < p and 2 < ps ≤ ps1 is constant for all s ≥ s1. Thus, let us call,pss≥s1 simply pjj≥1, where pj := psj .

Theorem 3.12. There exists a compact set Ks1 in R such thatMs1(t) ⊂ Ks1 , ∀ t ∈ R.

Proof. Multiplying the equation ups1(t) + C(t)|ups1

(t)|ps1−2ups1(t) = f (t, ups1

(t)) by ups1(t) and

using the Young’s Inequality we obtain

12

ddt|ups1

(t)|2 ≤ −α

2|ups1

(t)|ps1 + c, t ≥ τ

where c > 0 is a constant. Therefore, the map ys1(t) := |ups1(t)|2 satisfies the inequality

ddt

ys1(t) ≤ −α(ys1(t))ps1 /2 + 2c, t ≥ τ.

So, by Lemma 5.1 in [31],

|ups1(t)|2 ≤

(2cα

)2/ps1+

(α( ps1

2− 1)(t− τ)

)− 2ps1−2

, ∀ t ≥ τ.

Let ξ0 > 0 such that(α( ps1

2 − 1)ξ0)− 2

ps1−2 ≤ 1, then

|ups1(t)| ≤

[(2cα

)2/ps1

+ 1

]1/2

=: κs1 , ∀ t ≥ ξ0 + τ. (3.1)

Thus, consider the compact set in R defined by Ks1 := B(0, κs1).Consider now t ∈ R arbitrarily fixed and choose τ such that t− τ > ξ0. By the invariance of

the pullback attractor Vs1 , we have Ss1(t, τ)Ms1(τ) =Ms1

(t). So, given an arbitrary elementw ∈ Ms1

(t) we have that w = Ss1(t, τ)uτ with uτ ∈ Ms1(τ). Since that κs1 and ξ0 did not

depend on the initial data we have by (3.1) that w ∈ Ks1 . Therefore,Ms1(t) ⊂ Ks1 .

Consider from now on L < α where α is from Assumption C and the constant r0 = r0(M)

in Proposition 2.2 for M = Ks1 , where Ks1 is from Theorem 3.12. The set Ks1 is compact, inparticular bounded, so given ` ∈ R and δ0 there exists t0 = t0(`, δ0, Ks1) < ` such that

distR(S(`, t0)Ks1 ;A∞(`)) <δ0

4|Ω|1/2 , (3.2)

where S(`, t0)ut0 := u(`, ut0) is the solution of (2.2) and distR(S(`, t0)Ks1 ;A∞(`)) is the Haus-dorff semi-distance between S(`, t0)Ks1 and A∞(`) in R. Let ψ0 ∈ Ks1 be arbitrarily fixed.

Weak upper semicontinuity of pullback attractors 9

Let Sj(t, τ) be the process defined by problem (2.1) with the exponent parameter pj andconsider uj(t) := Sj(t, t0)ψ0.

Let us first prove the following three technical lemmas and then we present our mainresult.

Lemma 3.13. Given ` ∈ R and t0 ≤ ` as in (3.2), there exists a positive constant κ such that∣∣∣|uj+1(t)|pj−1 − |uj+1(t)|pj+1−1∣∣∣ ≤ κ|pj − pj+1|,

for all j ∈N and t ≥ t0.

Proof. By the Mean Value Theorem we conclude that∣∣∣|uj+1(t)|pj−1 − |uj+1(t)|pj+1−1∣∣∣ = ∣∣∣|uj+1(t)|θj ln |uj+1(t)|

∣∣∣ |pj − pj+1|,

for some θj ∈ (pj+1, pj). Consider the continuous function gθ : [0, r0]→ R given by

gθ(x) =

xθ ln x if x ∈ (0, r0]

0 if x = 0,

where r0 = r0(Ks1) is as in Proposition 2.2 and θ ≥ 1 is a given number. Using this continuousfunction defined in the compact set [0, r0] with θ = 2 when |uj+1(t)| < 1 and with θ = 2 + ε0

when |uj+1(t)| ≥ 1, there exists a positive constant κ such that∣∣∣|uj+1(t)|θ ln |uj+1(t)|∣∣∣ ≤ κ,

for all j ∈N and t ≥ t0 and the result follows.

Now, we can establish the following hypothesis

(H2) Given ` ∈ R and t0 ≤ ` as in (3.2), for each j ∈N,

|pj − pj+1| <[

δ20

52j Mκ2|Ω|e(2L+1)(`−t0)(`− t0)

]1/2

.

Lemma 3.14. Given ` ∈ R, consider t0 = t0(`) ≤ ` as in (3.2). If condition (H2) is fulfilled for agiven δ0 > 0, then

distR(Sj(`, t0)Ks1 ; Sj+1(`, t0)Ks1) ≤δ0

5j|Ω|1/2 ,

for all j ∈N.

Proof. Consider t0 ≤ ` < T. Subtracting the two equations in (2.1) and multiplying by uj(t)−uj+1(t), t ∈ [t0, T], we obtain

12

ddt|uj(t)− uj+1(t)|2 + C(t)[|uj(t)|pj−2uj(t)− |uj+1(t)|pj+1−2uj+1(t)][uj(t)− uj+1(t)]

= [ f (t, uj(t))− f (t, uj+1(t))][uj(t)− uj+1(t)].

10 J. Simsen

Adding ±C(t)|uj+1(t)|pj−2uj+1(t), using that f is Lipschitz with respect to the second variableand that for any ξ, η ∈ Rn, (|ξ|p−2ξ − |η|p−2η)(ξ − η) ≥ 0, we obtain

12

ddt|uj(t)− uj+1(t)|2 ≤ L|uj(t)− uj+1(t)|2

− C(t)[|uj+1(t)|pj−2 − |uj+1(t)|pj+1−2] uj+1(t)

(uj(t)− uj+1(t)

)≤ L|uj(t)− uj+1(t)|2 + M

∣∣∣|uj+1(t)|pj−1 − |uj+1(t)|pj+1−1∣∣∣ |uj(t)− uj+1(t)|

≤(

L +12

)|uj(t)− uj+1(t)|2 +

M2

∣∣∣|uj+1(t)|pj−1 − |uj+1(t)|pj+1−1∣∣∣2 ,

for all t ∈ [t0, T]. Using Lemma 3.13 we obtain

ddt|uj(t)− uj+1(t)|2 ≤ (2L + 1) |uj(t)− uj+1(t)|2 + Mκ2|pj − pj+1|2,

for all t ∈ [t0, T]. From condition (H2),

|pj − pj+1|2 <δ2

0

52j Mκ2|Ω|e(2L+1)(`−t0)(`− t0).

Then,

ddt|uj(t)− uj+1(t)|2 ≤ (2L + 1) |uj(t)− uj+1(t)|2 + Mκ2 δ2

0

52j Mκ2|Ω|e(2L+1)(`−t0)(`− t0),

for all t ∈ [t0, T]. Integrating from t0 to ` and using that uj(t0) = uj+1(t0) = ψ0, we obtain

|uj(`)− uj+1(`)|2 ≤δ2

0

52j|Ω|e(2L+1)(`−t0)+∫ `

t0

(2L + 1) |uj(t)− uj+1(t)|2dt.

So, by the Gronwall–Bellman Lemma we obtain

|uj(`)− uj+1(`)| ≤δ0

5j|Ω|1/2 ,

for all j ∈N. Thus,

distR(Sj(`, t0)ψ0; Sj+1(`, t0)Ks1) = infb∈Sj+1(`,t0)Ks1

distR(Sj(`, t0)ψ0; b)

≤ distR(Sj(`, t0)ψ0; Sj+1(`, t0)ψ0)

= |uj(`)− uj+1(`)| ≤δ0

5j|Ω|1/2 .

Since ψ0 ∈ Ms1 was arbitrary, we conclude that

distR(Sj(`, t0)Ks1 ; Sj+1(`, t0)Ks1) = supψ0∈Ks1

distR(Sj(`, t0)ψ0; Sj+1(`, t0)Ks1)

≤ δ0

5j|Ω|1/2 .

Lemma 3.15. Given ` ∈ R, consider t0 = t0(`) ≤ ` as in (3.2). Given δ0 > 0, we have

distR(Si(`, t0)Ks1 ; S(`, t0)Ks1) ≤δ0

4|Ω|1/2 ,

for i large enough.

Weak upper semicontinuity of pullback attractors 11

Proof. Let ψ0 ∈ Ks1 arbitrarily fixed. From Theorem 2.1,

|Si(`, t0)ψ0 − S(`, t0)ψ0| = |ui(`)− u(`)| < δ0

4|Ω|1/2 ,

for i large enough. So,

distR(Si(`, t0)ψ0; S(`, t0)Ks1) = infb∈S(`,t0)Ks1

distR(Si(`, t0)ψ0; b)

≤ distR(Si(`, t0)ψ0; S(`, t0)ψ0)

<δ0

4|Ω|1/2 .

Since ψ0 ∈ Ks1 was arbitrary, we conclude that

distR(Si(`, t0)Ks1 ; S(`, t0)Ks1) = supψ0∈Ks1

distR(Si(`, t0)ψ0; S(`, t0)Ks1) ≤δ0

4|Ω|1/2 ,

for i large enough.

Now, we are in conditions to establish the main result.

Theorem 3.16. Consider f = B|R×R : R×R → R with L < α (α is from Assumption C) andsuch that the limit problem (2.2) has a strongly bounded dissipative process. Assume condition (H). Ifcondition (H2) is fulfilled for a given δ0 > 0, then for each ` ∈ R,

As(`) ⊂ Oδ0(A∞(`)) = z ∈ H; infa∈A∞(`) ‖z− a‖H < δ0,

for s large enough.

Proof. Consider ` ∈ R arbitrarily fixed, t0 = t0(`) ≤ ` as in (3.2) and the sequence of func-tions ps(·)s∈N defined by p1(·) = p1(·), p2(·) = p2(·), . . . , ps1−1(·) = ps1−1(·), ps1(·) ≡ps1 , ps1+1(·) ≡ ps1 , . . . Applying Theorem 3.11 for this sequence of exponent functions and forthe original sequence of diffusion coefficients, we have that

dist(As(`);Ms1(`)) < δ0/4

for s large enough. Here dist(As(`);Ms1(`)) is the Hausdorff semi-distance between As(`)

andMs1(`) in the Hilbert space H. So,

dist(As(`);A∞(`)) ≤ dist(As(`);Ms1(`)) + dist(Ms1(`);A∞(`))

< δ0/4 + |Ω|1/2 distR(Ms1(`);A∞(`)), (3.3)

for s large enough.By the invariance of the pullback attractor Vs1 we have S1(`, t0)Ms1(t0)=Ms1(`). Then,

distR(Ms1(`);A∞(`)) = distR(S1(`, t0)Ms1(t0);A∞(`))

≤ distR(S1(`, t0)Ks1 ;A∞(`))

≤i

∑j=1

distR(Sj(`, t0)Ks1 ; Sj+1(`, t0)Ks1) (3.4)

+ distR(Si+1(`, t0)Ks1 ; S(`, t0)Ks1) + distR(S(`, t0)Ks1 ;A∞(`)),

for all i ∈N. Using (3.2), Lemma 3.14, Lemma 3.15 and letting i→ +∞ in (3.4), we obtain

distR(Ms1(`);A∞(`)) <+∞

∑j=1

δ0

5j|Ω|1/2 +δ0

4|Ω|1/2 +δ0

4|Ω|1/2 =3δ0

4|Ω|1/2 . (3.5)

Using (3.5) in (3.3) the result follows.

12 J. Simsen

4 Final remarks

Comparing this nonautonomous problem with the previous autonomous in [29], a naturalquestion that raise is it also possible for each given ` ∈ R, for large s, to include the sectionAs(`) of the pullback attractors Us of problem (1.1) into a neighborhood of an interval? UsingTheorem 3.16, this will be true if it is possible to prove that A∞(`) is included into an intervalor it is just one equilibrium point of the limit problem (2.2).

Acknowledgements

J. Simsen has been partially supported by FAPEMIG (Brazil) - processes PPM 00329-16.

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