Robustness with respect to exponents for nonautonomous reaction–diffusion equations

Robustness with respect to exponents for nonautonomous reaction–diffusion equations

Rodrigo A. Samprogna

and Jacson Simsen

Instituto de Matemática e Computação, Universidade Federal de Itajubá, Av. BPS n. 1303, Bairro Pinheirinho, Itajubá 37500-903, Brazil Received 23 November 2017, appeared 14 February 2018

Communicated by Christian Pötzsche

Abstract. In this work we consider a family of nonautonomous problems with homoge- neous Neumann boundary conditions and spatially variable exponents with equation of the form

∂u_λ

∂t

(t)−div

D(t)|∇u_λ(t)|^pλ(x)−2∇u_λ(t)+|u_λ(t)|^pλ(x)−2u_λ(t) =B(t,u_λ(t)). We study the continuity of the flow and we study the behavior of attractors when p_λ(·)→p(·)inL^∞(_Ω)asλ→_∞whereΩis a bounded smooth domain inR^N. Keywords: p(x)-Laplacian, variable exponents, pullback attractor and nonautonomous asymptotic behavior.

2010 Mathematics Subject Classification: 35K92, 35K55, 35B41, 35B40, 37B55.

1 Introduction

In several physical, chemical and biological problems the reaction–diffusion systems can be a good model to describe the behavior of the problem, and in many of these problems may appear operators in which some exponent pdepends on the spatial variable, as in the case of the operator p(x)-Laplacian [3,4,9]. These require the use of function spaces with spatially dependent exponents and new mathematical techniques.

The asymptotic behavior of nonautonomous evolution problems has been investigated recently [6,8]. An interesting problem is to investigate how is the asymptotic behavior of these problems with the variation of parameters, more specifically, try to establish the existence of attractors for each parameter and to study the continuity of these attractors with respect to the variation of the parameters, see [6,12,13].

In this paper we establish upper semicontinuity of pullback attractors for a nonautonom- ous evolution equation of the form







∂uλ

∂t (t)−div

D(t)|∇u_λ(t)|^pλ(x)−2∇u_λ(t)+|u_λ(t)|^pλ(x)−2u_λ(t) =B(t,u_λ(t)),

uλ(τ) =u_0λ (P_λ)

BCorresponding author. Email: samprogna@hotmail.com

with a homogeneous Neumann boundary condition, for (t,x) ∈ (τ,+_∞)×_Ω where Ω is a bounded smooth domain inR^N for someN ≥1 and the initial conditionuλ(τ)∈ H := L²(_Ω). The terms p_λ,BandDare assumed to satisfy:

Assumpition pp_λ(·)∈C(_Ω,R), for eachλ∈[0,∞), satisfies (P1) there are m,M ∈_Rsuch that

2<m≤ p⁻_λ :=min_x∈Ωp_λ(x)≤ p⁺_λ :=max_x∈Ωp_λ(x)≤ M;

(P2) p_λ → pin L^∞(_Ω)for some psuch that p(·)∈C(_Ω,R)andm≤ p⁻ ≤ p⁺ ≤ M.

Assumpition BThe mappingB:[_τ,T]×H→His such that (B1) there exists L≥0 such that

kB(t,x₁)−B(t,x₂)k_H ≤ Lkx₁−x₂k_H for allt∈ [τ,T]andx₁,x₂ ∈ H;

(B2) for allx∈ Hthe mapping t→B(t,x)_{belongs to} L²(_τ,T;H)_;

(B3) the function t → kB(t, 0)k_H is nondecreasing, absolutely continuous and bounded on compact subsets ofR.

Assumpition DD:[τ,T]×_Ω→_Ris a function in L^∞([τ,T]×_Ω)such that

(D1) there are positive constants, β and M such that 0 < β ≤ D(t,x) ≤ M for almost all (t,x)∈[τ,T]×_Ω;

(D2) D(t,x)≥D(s,x)for eachx ∈_Ωandt ≤sin[τ,T].

The authors in [6] also considered the nonautonomous problem and proved the robustness with respect to the diffusion coefficient whereas in this work we study the robustness with respect to the exponents.

The paper is organized as follows. In Section 2 we present the time-dependent evolution operator and the results of [6] that ensure some properties of the operator and existence and uniqueness of solution for the problem. The work [6] also ensures the existence of a pullback attractor for the problem, these results are in Section 3. With the objective of guaranteeing the upper semicontinuity of attractors we need to develop some new uniform estimates for the problem, these estimates are established in Section 4. In Section 5, we ensure the continuity of the process associated with the problem and the upper semicontinuity of the pullback attractors.

2 Preliminaries

In this section we present some definitions about Lebesgue and Sobolev spaces with variable exponents, a general theory about this spaces can be found in [4,5,11]. Also we show some re- sults about the operator associated with our problem and the results that ensure the existence and uniqueness of solution, following [6].

Let us recall the definitions of the Lebesgue and Sobolev spaces with variable exponents.

Considering p∈ L^∞₊(_Ω):= {q∈ L^∞(_Ω): ess infq≥1}, then L^p(x)(_Ω):=

u :Ω→_R:uis mensurable, Z

Ω|u(x)|^p(x)dx<_∞

is a Banach space with the normkuk_p(x):=inf{λ>0 :ρ ^u_λ

≤1}, whereρ(u):=R

Ω|u(x)|^p(x)dx.

Furthermore,

W^1,p(x)(_Ω):=ⁿu∈ L^p(x)(_Ω);|∇u| ∈ L^p(x)(_Ω)^o which is a Banach space with the norm kuk_W1,p(x)(_Ω) := kuk_p(x)+k∇uk_p(x).

The authors in [6] considered, for eachλ∈[0,∞)andt∈R, the operatorAλ(t):Xλ →X_λ^∗, where X_λ :=W^1,pλ(x)(_Ω)with normk · k_Xλ := k · k_W1,pλ(x)(_Ω), defined by

A_λ(t)u(v):=

Z

ΩD(t,x)|∇u(x)|^pλ(x)−2∇u(x)· ∇v(x)dx+

Z

Ω|u(x)|^pλ(x)−2u(x)v(x)dx, for eachu,v∈ X_λand they have proved that the operator A_λ(t)is monotone, hemicontinuous and coercive for each t ∈ [_τ,T] _and λ ∈ [_0,∞). Then, they concluded that the operator is maximal monotone and the realization operator of A_λ(t)atH=L²(_Ω)is maximal monotone in H, for eacht ∈[τ,T]andλ∈[0,∞). With this it is possible to show that the operatorA_λ(t) is the subdifferential∂ϕ_λ(t)of the convex, proper and lower semicontinuous mapϕ_λ(t)_:H→ R∪ {+_∞}given by

ϕ_λ(t)(u)_:= (R

Ω D(t,x)

p_λ(x)|∇u|^pλ(x)dx+R

Ω 1

p_λ(x)|u|^pλ(x)dx, ifu∈ X_λ

+_∞, otherwise. (2.1)

Note that, defining A(t) as A_λ(t) with function p in place of function p_λ we have that all properties described above holds for operator A(t)_{, for all}t∈ [_τ,T], we will denote the space X:=W^1,p(x)(_Ω).

We will present some estimates that will be useful in the course of the work.

Theorem 2.1([5]). If u∈ L^p(x)(_Ω). Then

i) kuk_p(x) <1(=1;>1)if and only ifρ(u)<1(=1;>1); ii) ifkuk_p(x) >1, thenkuk^p−

p(x) ≤ρ(u)≤ kuk^p+

p(x); iii) ifkuk_p(x) <1, thenkuk^p+

p(x) ≤ρ(u)≤ kuk^p−

p(x).

From Lemmas 2.2 and 2.3 in [6] and Proposition 3.1 in [1] we can conclude the following result.

Lemma 2.2. For t∈ [τ,T], we have that for every u∈X

hA(t)u,ui_X∗,X≥







min{1,β}

2^p+ kuk^p_X⁺, ifkuk_X<1

min{1,β}

2^p+ kuk^p_X⁻, ifkuk_X≥1

Remark 2.3. It is obvious that both of the above results are satisfied if instead of function p we take function p_λ for each λ ∈ [0,∞)and with their respective spaces X_λ and bounds p⁻_λ and p⁺_λ.

We recall the result of existence of solution from [6].

Theorem 2.4. If B : [τ,T]×H → H satisfies Assumptions (B1) and (B2) and u_0λ ∈ H, then for each λ ∈ [_0,∞) there exists a unique strong global solution of the problem (P_λ), i.e., there exists u_λ ∈C([τ,T];H), with u_λ(τ) =u_0λ such that

du_λ

dt (t) +A_λ(t)u_λ(t) =B(t,u_λ(t)) a.e. on[τ,T].

3 Pullback attractor

The theory about pullback attractors can be found in [2]. The existence of pullback attractor for the Problem (P_λ) was ensured in [6], for eachλ∈[0,∞).

Definition 3.1. An evolution process in a metric space X is a family {U(t,τ) : X → X; t ≥ τ∈_R}satisfying:

i) U(τ,τ) =Id_X;

ii) U(t,τ) =U(t,s)U(s,τ),τ≤s ≤t.

Definition 3.2. Let{U(t,τ); t ≥ τ ∈ _R}be an evolution process in a metric space X. Given AandBsubsets of X, we say that Apullback attractsBat timet if

lim

τ→−_∞dist_H(U(t,τ)B,A) =0, where distH denote the Hausdorff semi-distance.

Definition 3.3. A family of subsets {A(t): t ∈ _R}of X is called a pullback attractor for the evolution process {U(t,τ); t ≥ τ ∈ _R} if, for each t ∈ _R, A(t) is compact, A(t) pullback attracts all bounded subsets ofXat timetand the family is invariant, i.e.,U(t,τ)A(τ) = A(t) for anyt≥ τ.

Note that, for eachλ∈[0,∞), Theorem2.4defines an evolution process {U_λ(t,τ):t ≥τ} in the space H associated with problem (P_λ). Indeed, given u_0λ ∈ H define U_λ(t,τ)u_0λ := u_λ(t) where u_λ is a solution of problem (P_λ) with initial condition u_λ(τ) = u_0λ, see [6] for details. Denote by {U(t,τ) : t ≥ τ} the evolution process associated with a problem like Problem (P_λ) but with the function pin place of p_λ.

The next result was developed in [6] and ensures, for each λ ∈ [0,∞), the existence of pullback attractors.

Theorem 3.4. The evolution process associated with Problem (P_λ) has a pullback attractor A_λ = {A_λ(t): t∈_R}.

This theorem also ensure the existence of a pullback attractor A= {A(t) _:t ∈ _R}_{for the} problem with functionp.

4 Estimates

Our objective in this work is to show the upper semicontinuity of the pullback attractors, for this, we will need develop some estimates uniform inλ.

Theorem 4.1. Let u_λ be a solution of Problem(P_λ). Then there exist a constant T₁ ≥ 0 and a non decreasing function B₁:R→_Rsuch that

ku_λ(t)k_H ≤B₁(t), ∀ t≥ T₁+τ andλ∈ [0,∞).

Proof. Multiplying the equation of the Problem (P_λ) byu_λ(t), we obtain 1

2 d

dtku_λ(t)k²_H+hA_λ(t)u_λ(t),u_λ(t)i=hB(t,u_λ(t)),u_λ(t)i.

It is easy to see thatku_λ(t)k_H ≤4(|_Ω|+1)²ku_λ(t)k_Xλ. Without loss of generality assume that ku_λ(t)k_X

λ ≥ 1, if not the theorem is already proved. Then by, Lemma 2.2 and the Cauchy–

Schwarz inequality, we obtain 1

2 d

dtku_λ(t)k²_H ≤ −^min{1,β}

2^p+λ ku_λ(t)k^p_X^−λ

λ +hB(t,u_λ(t))−B(t, 0),u_λ(t)i+hB(t, 0),u_λ(t)i

≤ −^min{_1,β}

2^M ku_λ(t)k^m_X

λ +C₁ku_λ(t)k²_X

λ+C₂(t)ku_λ(t)k_Xλ whereC₁ := L

4(|_Ω|+1)²²andC₂(t):=4(|_Ω|+1)²kB(t, 0)k_H.

Ifθ:= ^m₂, ¹_θ +_θ¹0 =1 and _m¹ + _m¹0 =1. Then from Young’s inequality with ε>0, we obtain C₁ku_λ(t)k²_X

λ +C₂(t)ku_λ(t)k_Xλ = ^C1ε

ε ku_λ(t)k²_X

λ+ ^C2(t)ε

ε ku_λ(t)k_Xλ

≤ ¹ θ⁰

C₁ ε

θ⁰

+ ¹

θε^θku_λ(t)k^m_X

λ+ ¹

m⁰

C₂(t) ε

m⁰

+ ¹

mε^mku_λ(t)k^m_X

λ. Chooseε₀ >0 such that

γ:= ^min{1,β} 2^M −¹

θε^θ₀− ¹

mε^m₀ >0, we have

1 2

d

dtku_λ(t)k²_H+γku_λ(t)k^m_X

λ ≤ ¹

θ⁰ C₁

ε₀ θ⁰

+ ¹ m⁰

C2(t) ε₀

m⁰

. Letδ(t):= ²

θ⁰ C1

ε0

θ⁰

+_m²0 C2(t) ε0

m⁰

, ˜γ:= _[ ^2γ

4(|_Ω|+1)²]^m andy_λ(t):=ku_λ(t)k²_H. Then y⁰_λ(t) +γy˜ _λ(t)^m2 ≤δ(t), ∀t ≥τ.

From a slight generalization of Lemma 5.1 in [14], we obtain y_λ(t)≤

δ(t)

˜ γ

_m² +

˜ γ

m−2 2

(t−τ) −_m²₋₂

. Let T₁>0 such that

˜ γ ^m−₂²

T₁−_m²₋₂

≤1. Then ku_λ(t)k_H ≤

δ(t)

˜ γ

_m¹

+1=:K₁(t),

for all t≥T₁+τ. Observe thatK₁(t)is nondecreasing by Assumption (B3).

TakingB₁(t)_:=_max{K₁(t)_{, 4}(|_Ω|+₁)²}the theorem follows.

Theorem 4.2. Given T> τand a bounded set B⊂ H, there exists D₁(T)>0such thatku_λ(t)k_H ≤ D₁(T), for allτ≤t ≤T andλ∈ [0,∞)such that u_0λ ∈B.

Proof. Without loss of generality assume that ku_λ(t)k_Xλ ≥ 1, if not the theorem is already proved. Proceeding as in the first lines of the proof of the previous theorem above, we obtain

1 2

d

dtku_λ(t)k²_H ≤ −γku_λ(t)k^m_X

λ+ ¹ θ⁰

C₁ ε₀

θ⁰

+ ¹ m⁰

C2(t) ε₀

m⁰

≤ ¹ θ⁰

C₁ ε₀

θ⁰

+ ¹ m⁰

C₂(t) ε₀

m⁰

whereε₀>0 was given in the proof of the previous theorem. Integrating fromτtot≤ T, we obtain

ku_λ(t)k²_H ≤ ku_λ(τ)k²_H+ (t−τ)

"

1 θ⁰

C₁ ε₀

θ⁰#

+K(t)

whereK(t):= ¹

m⁰ε^m₀⁰

Rt

τC₂(s)^m0ds<_∞is bounded.

Indeed, as m > 2 we have _m¹ < ¹₂, therefore ¹₂ < _m¹0, and then 1 < m⁰ < 2, because

1

m +_m¹0 = 1. Remembering the definition of C2(t) and the assumption (B2), we can see that K(T)is bounded.

Consequently

ku_λ(t)k²_H ≤ ku_λ(τ)k²_H+ (T−τ)

"

1 θ⁰

C₁ ε₀

θ⁰#

+K(T).

Theorem 4.3. Let u_λ ∈ C([τ,∞);H) be the global solution of Problem (P_λ). Then there exist a constant T₂ >0and a nondecreasing function B₂ :R→_Rsuch that

ku_λ(t)k_X

λ ≤B₂(t), ∀t ≥T₂+τ, λ∈[0,∞). Proof. Letu_λ be the global solution of (P_λ). Using the identity

d

dtϕp_λ(t)(u_λ(t)) =

∂ϕp_λ(t)(u_λ(t)),duλ

dt (t)

=

A_λ(t)u_λ(t),du_λ dt (t)

=

B(t,u_λ(t))−^duλ

dt (t),du_λ dt (t)

= −

B(t,u_λ(t))−^duλ dt (t)

2 H

+

B(t,u_λ(t))− ^duλ

dt (t),B(t,u_λ(t))

≤ −

B(t,u_λ(t))−^duλ dt (t)

2 H

+ ¹ 2

B(t,u_λ(t))− ^duλ dt (t)

2

H

+ ¹

2kB(t,u_λ(t))k²_H. Therefore,

d

dtϕ_pλ(t)(u_λ(t)) + ¹ 2

B(t,u_λ(t))−^duλ dt (t)

2 H

≤ ¹

2kB(t,u_λ(t))k²_H,

and thus,

d

dtϕ_pλ(t)(uλ(t))≤ ¹ 2

kB(t,uλ(t))−B(t, 0)k_H+kB(t, 0)k_H 2

≤ ¹ 2

Lku_λ(t)k_H+kB(t, 0)k_H 2

. From Theorem4.1, we obtain

d

dtϕ_pλ(t)(u_λ(t))≤ M₁(t), ∀t ≥T₁+τ, where M₁(t):= ¹₂[LB₁(t) +kB(t, 0)k_H]².

From the definition of subdifferential, we have

ϕ_pλ(t)(u_λ(t))≤∂ϕ_pλ(t)(u_λ(t))_,u_λ(t)_. Thus,

1 2

d

dtku_λ(t)k²_H+ϕp_λ(t)(u_λ(t))≤ du_λ

dt (t),u_λ(t)

+∂ϕp_λ(t)(u_λ(t)),u_λ(t)

= du_λ

dt (t) +∂ϕ_pλ(t)(u_λ(t)),u_λ(t)

=hB(t,u_λ(t))_,u_λ(t)i

≤ kB(t,u_λ(t))k_Hku_λ(t)k_H

≤ ¹

2kB(t,u_λ(t))k²_H+¹

2ku_λ(t)k²_H

≤ M₁(t) +¹

2B₁(t)², ∀t≥ T₁+τ.

(4.1)

Fixingr >0 and integrating both sides of (4.1) over(t,t+r)fort≥ T₁+τ, Z _t+r

t ϕ_pλ(s)(u_λ(s))ds≤ ¹

2ku_λ(t)k²_H+

Z _t+r

t M₁(s) +¹

2B₁(s)²ds

≤ ¹

2B₁(t)²+

Z _t+r

t M₁(s) +¹

2B₁(s)²ds=_:a₃(t)_, Lety_λ(s) = ϕp_λ(s)(u_λ(s)),g :=0 andh(s):= M₁(s). Then

Z _t+r

t g(s)ds=0=:a₁(t),

Z _t+r

t h(s)ds=:a2(t),

Z _t+r

t yλ(s)ds≤a3(t), from a slight generalization of the uniform Gronwall lemma [14], we obtain

y_λ(t+r)≤

a₃(t)

r +a₂(t)

e⁰ =: ˜r₁(t), ∀ t≥ T₁+τ. (4.2) Therefore,

Z

Ω

D(t,x)

p_λ(x) |∇u_λ(`,x)|^pλ(x)dx+

Z

Ω

1

p_λ(x)|u_λ(`,x)|^pλ(x)dx≤r˜₁(t),

for all`≥T₁+τ+randλ∈[0,∞). Then min{1,β}

M [ρ_λ(∇u_λ(`)) +ρ<sub>λ</sub>(u<sub>λ</sub>(`))]≤r˜₁(t) for all`≥T₁+τ+randλ∈[0,∞), and hence,

ρ_λ(∇u_λ(`)) +ρ<sub>λ</sub>(u<sub>λ</sub>(`))≤ ^M

min{1,β}^r˜1(t) (4.3) for all`≥T₁+τ+randλ∈[0,∞).

If`≥T<sub>1</sub>+τ+randku<sub>λ</sub>(`)k_X

λ ≥1 there are four cases to analyze.

Case 1: Ifk∇u_λ(`)k_p

λ(x)≥1 andku_λ(`)k_p

λ(x)≥1 we know that k∇u_λ(`)k^p_p^−λ

λ(x)≤ ρ_λ(∇u_λ(`))≤ k∇u<sub>λ</sub>(`)k^p

+ λ

p_λ(x)

and

kuλ(`)k^p_p^−λ

λ(x) ≤ρ_λ(_uλ(`))≤ kuλ(`)k^p

+ λ

p_λ(x). Sincem≤ p⁻_λ ≤ p⁺_λ ≤ M and using (4.3), we have

ku_λ(`)k_Xλ ≤R₁(t), t ≥T₂+τ, λ∈[0,∞), whereR₁(t):=2 _M

min{1,β}r˜₁(t)^m1 andT₂:=T₁+r.

Case 2: Ifk∇u_λ(`)k_p

λ(x)≥1 andku_λ(`)k_p

λ(x)≤1 we know that k∇u_λ(`)k^p−λ

p_λ(x)≤ ρ_λ(∇u_λ(`))≤ k∇u<sub>λ</sub>(`)k^p+λ

p_λ(x)

and

ku_λ(`)k^p+λ

pλ(x) ≤ρ_λ(u_λ(`))≤ ku<sub>λ</sub>(`)k^p−λ

pλ(x). Sincem≤ p⁻_λ ≤ p⁺_λ ≤ M and using (4.3), we have

ku_λ(`)k_X

λ ≤R₂(t), t≥T₂+τ, λ∈ [0,∞), whereR₂(t):=_min^M_{1,β}r˜₁(t)^m1 +_min^M_{1,β}r˜₁(t)^M1 .

Case 3: Ifk∇u_λ(`)k_p

λ(x)≤1 andku_λ(`)k_p

λ(x)≥1 we know that k∇u_λ(`)k^p

+ λ

p_λ(x)≤ ρ_λ(∇u_λ(`))≤ k∇u<sub>λ</sub>(`)k^p_p^−λ

λ(x)

and

ku_λ(`)k^p

− λ

p_λ(x) ≤ρ_λ(u_λ(`))≤ ku<sub>λ</sub>(`)k^p

+ λ

p_λ(x). Sincem≤ p⁻_λ ≤ p⁺_λ ≤ M and using (4.3), we have

ku_λ(`)k_Xλ ≤R₃(t), t≥T₂+τ, λ∈ [0,∞), whereR₃(t)_:= R₂(t)_.

Case 4: Ifk∇u_λ(`)k_p

λ(x)≤1 andku_λ(`)k_p

λ(x)≤1 we know that k∇u_λ(`)k^p+λ

pλ(x)≤ ρ_λ(∇u_λ(`))≤ k∇u<sub>λ</sub>(`)k^p−λ

pλ(x)

and

kuλ(`)k^p

+ λ

p_λ(x)≤ρ_λ(uλ(`))≤ kuλ(`)k^p_p^−λ

λ(x). Sincem≤ p⁻_λ ≤ p⁺_λ ≤ Mand using (4.3), we have

ku_λ(`)k_Xλ ≤ R₄(t), t ≥T₂+τ, λ∈[0,∞), where R₄(t):=2 _M

min{1,β}r˜₁(t)^M1 . In summary, defining

B2(t):=max (

1, 2

"

M

min{1,β}^r˜1(t) _m¹

+

M

min{1,β}^r˜1(t) _M¹#)

we have

ku_λ(t)k_Xλ ≤ B₂(t), t≥ T₂+τ, λ∈ [0,∞).

Corollary 4.4. Let T₂ >0obtained in the Theorem4.3. The following statements are satisfied.

a) Let u_λ be a solution of the Problem (P_λ) in [τ,∞). There exist a nondecreasing function B₃ : R→_Rsuch that

ku_λ(t)k_Xm ≤ B₃(t), ∀t ≥T₂+τandλ∈[0,∞), where Xm =W^1,m(_Ω);

b) There exist a family of bounded sets D:= D(t)_t∈Rin X_msuch thatA_λ(t)⊂ D(t)for each t and λ∈ [0,∞), whereA_λ is the pullback attractor for the evolution process associated with Problem (P_λ);

c) ∪_λ∈[0,∞)A_λ(t)is compact in H for each t∈ _R.

Proof. The item a) follows from Theorem4.3.

The item b) follows from item a).

The item c) follows from compact embedding ofX_m in H.

Theorem 4.5. Let u_λ be a solution of Problem(P_λ) such that u_λ(τ) = u_0λ ∈ X_λ and suppose that there is C>0such thatku_0λk_Xλ ≤C for allλ∈[0,∞). Given T> τ, then we have that there exists D₂(T)>₀such thatku_λ(t)k_X

λ ≤ D₂(T), for allτ≤ t≤T andλ∈[_τ,∞).

Proof. Proceeding as in the first lines of the proof of theorem above we obtain by Theorem4.1 that

d

dtϕp_λ(t)(u_λ(t))≤ ¹ 2

kB(t,u_λ(t))−B(t, 0)k_H+kB(t, 0)k_H 2

≤ ¹ 2

Lku_λ(t)k_H+kB(t, 0)k_H 2

≤ ¹ 2

LD₁(T) +kB(t, 0)k_H 2

. Integrating in(τ,t),t ≤T, we have

ϕp_λ(t)(u_λ(t))≤ ϕp_λ(τ)(u_λ(τ)) + ^L

2

2 D₁(T)²(T−τ) + ¹

2 Z _t

τ

kB(s, 0)k²_Hds+LD₁(T)

Z _t

τ

kB(s, 0)k_Hds.

Asku_λ(τ)k_X

λ ≤Cwe have that ϕ_pλ(τ)(u_λ(τ))≤C. Since^˜ min(1,β)

M [ρ(∇u_λ(t)) +ρ(u_λ(t))]≤ ϕp_λ(t)(u_λ(t)),

we can use assumptions (B2) and (B3) and divide in cases as in the proof of the above theorem and conclude the result.

5 Upper semicontinuity of pullback attractors

Finally we will show the upper semicontinuity of the pullback attractors but first we es- tablished in the next theorem the continuity of the process, this proofs the robustness of the problem and help us to proof our main result which is the upper semicontinuity of the pullback attractors.

Theorem 5.1. Let {U_λ(t,τ) : t ≥ τ ∈ _R}be the evolution process generated by the problem (P_λ).

If each ku_0λk_Xλ ≤ C and u_0λ → u0 in H when λ → _{∞, then Uλ}(t,τ)u_0λ → U(t,τ)u0 in H as λ→∞, uniformly for t in compact subsets ofR.

Proof. Subtracting equation (P_λ) from the limit equation gives d

dt(u_λ(t)−u(t)) +A_λ(t)u_λ(t)−A(t)u(t) =B(t,u_λ(t))−B(t,u(t)) for a.e. t∈[τ,T]. Then multiplying byu_λ(t)−u(t), we obtain

1 2

d

dtku_λ(t)−u(t)k²_H+hA_λ(t)u_λ(t)−A(t)u(t),u_λ(t)−u(t)i

=hB(t,u_λ(t))−B(t,u(t)),u_λ(t)−u(t)i

≤ kB(t,u_λ(t))−B(t,u(t))k_Hku_λ(t)−u(t)k_H ≤ Lku_λ(t)−u(t)k²_H

Moreover, for anyξ,η∈_R^N we have the following inequality for a constant p≥2 (see [5]):

|ξ|^p−2ξ− |η|^p−2η

·(ξ−η)≥ 1

2 p

|ξ−η|^p. (5.1)

Using (5.1), after some computation we obtain hA_λ(t)u_λ(t)−A(t)u(t),u_λ(t)−u(t)i

≥β Z

Ω

|∇u_λ(t)|^pλ(x)−2∇u_λ(t)− |∇u(t)|^pλ(x)−2∇u(t)(∇u_λ(t)− ∇u(t))dx +

Z

Ω

|u_λ(t)|^pλ(x)−2u_λ(t)− |u(t)|^pλ(x)−2u(t)(u_λ(t)−u(t))dx +β

Z

Ω

|∇u(_t)|^pλ(x)−2∇u(_t)− |∇u(_t)|^p(x)−2∇u(_t)(∇uλ(_t)− ∇u(_t))_dx +

Z

Ω

|u(t)|^pλ(x)−2u(t)− |u(t)|^p(x)−2u(t)(u_λ(t)−u(t))dx

≥β 1

2 MZ

Ω|∇u_λ(t)− ∇u(t)|^pλ(x)dx+ 1

2 MZ

Ω|u_λ(t)−u(t)|^pλ(x)dx +β

Z

Ω

|∇u(t)|^pλ(x)−2∇u(t)− |∇u(t)|^p(x)−2∇u(t)(∇u_λ(t)− ∇u(t))dx +

Z

Ω

|u(_t)|^pλ(x)−2u(_t)− |u(_t)|^p(x)−2u(_t)(uλ(_t)−u(_t))_dx.

Then 1 2

d

dtku_λ(t)−u(t)k²_H

≤Lkuλ(t)−u(t)k²_H

−β Z

Ω

|∇u(t)|^pλ(x)−2∇u(t)− |∇u(t)|^p(x)−2∇u(t)(∇u_λ(t)− ∇u(t))dx

−

Z

Ω

|u(t)|^pλ(x)−2u(t)− |u(t)|^p(x)−2u(t)(u_λ(t)−u(t))dx

=Lku_λ(t)−u(t)k²_H

−β Z

Ω

|∇u(t)|^pλ(x)−2− |∇u(t)|^p(x)−2∇u(t) (∇u_λ(t)− ∇u(t))dx

−

Z

Ω

|u(t)|^pλ(x)−2− |u(t)|^p(x)−2u(t) (u_λ(t)−u(t))dx

≤Lku_λ(t)−u(t)k²_H +β

Z

Ω

|∇u(t)|^pλ(x)−1− |∇u(t)|^p(x)−1|∇u_λ(t)− ∇u(t)|dx +

Z

Ω

|u(t)|^pλ(x)−1+|u(t)|^p(x)−1|u_λ(t)−u(t)|dx, a.e. in (τ,T).

Now, let us estimate the term Z

Ω

|∇u(t)|^pλ(x)−1− |∇u(t)|^p(x)−1|∇u_λ(t)− ∇u(t)|dx.

From Theorems 4.3 and 4.5 there exists a constant K := K(T), which is independent of λ, satisfying |∇u(t)| ≤ K for all t ∈ [τ,T]and a.e. for x ∈ Ω. By the mean value theorem, for each x ∈ _Ωand λ∈ [0,∞)there isq ∈ (p(x),p_λ(x)), ifp(x)≤ p_λ(x)(orq∈ (p_λ(x),p(x)), if p_λ(x)≤ p(x)) such that

|∇u(t)|^pλ(x)−1− |∇u(t)|^p(x)−1=

|∇u(t)|^q−1ln|∇u(t)||p_λ(x)−p(x)|

provided thatu(t)6=0. From the bound of|∇u(t)|and|∇u_λ(t)|we have that there isκ₁such that

|∇u(t)|^q−1ln|∇u(t)||∇u_λ(t)− ∇u(t)| ≤κ₁, for all t∈[τ,T]with u(t)6=0 and a.e.x∈_{Ω. Thus,}

|∇u(t)|^pλ(x)−1− |∇u(t)|^p(x)−1|∇u_λ(t)− ∇u(t)| ≤κ₁|p_λ(x)−p(x)|

for all t∈[τ,T]and a.e.x ∈_Ω.

Analogously, we can getκ₂ such that

|u(t)|^pλ(x)−1− |u(t)|^p(x)−1|u_λ(t)−u(t)| ≤κ₂|p_λ(x)−p(x)|

for all t∈[_τ,T]_{and a.e.}x ∈_Ω.

Therefore, 1

2 d

dtku_λ(t)−u(t)k²_H ≤ Lku_λ(t)−u(t)k²_H +β

Z

Ω

|∇u(t)|^pλ(x)−1− |∇u(t)|^p(x)−1|∇u_λ(t)− ∇u(t)|dx +

Z

Ω

|u(t)|^pλ(x)−1+|u(t)|^p(x)−1|u_λ(t)−u(t)|dx,

≤ Lku_λ(t)−u(t)k²_H

+κ₁kp_λ(x)−p(x)k_L∞(_Ω) Z

Ω|∇u_λ(t)− ∇u(t)|dx +κ₂kp_λ(x)−p(x)k_L∞(_Ω)

Z

Ω|u_λ(t)−u(t)|dx

≤ Lku_λ(t)−u(t)k²_H

+κ₁kp_λ(x)−p(x)k_L∞(_Ω)

1

2|_Ω|+¹

2k∇u_λ(t)− ∇u(t)k²_H

+κ₂kp_λ(x)−p(x)k_L∞(_Ω)

1

2|_Ω|+¹

2ku_λ(t)−u(t)k²_H

≤ Lku_λ(t)−u(t)k²_H

+κ₁kp_λ(x)−p(x)k_L∞(_Ω)

1

2|_Ω|+¹ 2K˜(T)

+κ₂kp_λ(x)−p(x)k_L∞(_Ω)

1

2|_Ω|+¹

2ku_λ(t)−u(t)k²_H

a.e. in[τ,T], by Theorems4.3and4.5.

Takeκ :=2κ₁+2κ₂+2|_Ω|+K^˜(T). Integrating fromτtot,t ≤T, we obtain ku_λ(t)−u(t)k²_H ≤ ku_0λ−u₀k²_H+ (t−τ)κkp_λ(x)−p(x)k_L∞(_Ω)

Z _t

τ

2L+κ₂kp_λ(x)−p(x)k_L∞(_Ω)

ku_λ(s)−u(s)k²_Hds Then, from the Gronwall–Bellman lemma, we obtain

ku_λ(t)−u(t)k²_H ≤ku_0λ−u₀k²_H+ (T−τ)κkp_λ(x)−p(x)k_L∞(_Ω)

e^{L(λ)(T−τ)} (5.2) for allt ∈ [τ,T], where L(λ) := 2L+κ₂kp_λ(x)−p(x)k_L∞(_Ω) is bounded. Therefore,u_λ → u inC([_τ,T]_;H)_asλ→_∞.

Theorem 5.2. The family of pullback attractors{A_λ(t):t ∈_R},λ∈[0,∞)is upper semicontinuous.

Proof. We will prove that for eacht∈_R,

dist(A_λ(t),A(t))→0, as λ→_∞.

Givent∈_Randε>0, letτ∈_Rbe such that

dist(U(t,τ)D(τ),A(t))< ^ε 3,

where∪_λ∈[0,∞)A_λ(τ)⊂ D(τ)_andD(τ)is a bounded set inH(see Corollary4.4).