Uniform attractors of non-autonomous Kirchhoff wave models

TL;DR

This paper proves the existence and stability of uniform attractors for a class of non-autonomous Kirchhoff wave equations with strong damping and supercritical nonlinearities, showing their behavior under perturbations.

Contribution

It establishes the existence and upper semicontinuity of uniform attractors for perturbed Kirchhoff wave models with supercritical growth nonlinearities.

Findings

Existence of compact uniform attractors for each perturbation parameter.

Upper semicontinuity of attractors with respect to the perturbation parameter.

Applicability to supercritical nonlinear growth conditions.

Abstract

The paper investigates the existence and upper semicontinuity of uniform attractors of the perturbed non-autonomous Kirchhoff wave equations with strong damping and supercritical nonlinearity: $u_{tt}-\Delta u_{t}-(1+\epsilon\|\nabla u\|^{2})\Delta u+f(u)=g(x,t)$, where $\epsilon\in [0,1]$ is a perturbed parameter. It shows that when the nonlinearity $f(u)$ is of supercritical growth $p: \frac{N+2}{N-2}=p^*<p<p^{**}=\frac{N+4}{(N-4)^+}$: (i) the related evolution process has a compact uniform attractor $\mathcal{A}_\ls^\e $ for each $\epsilon\in [0,1]$; (ii) the family of uniform attractor $\mathcal{A}_\ls^\e $ is upper semicontinuous on the perturbed parameter $\epsilon$ in the sense of partially strong topology.

Equations

$$u_{tt}-\Delta u_t-(1+ϵ\Vert \nabla u{\Vert }^2)\Delta u+f(u)=g(x,t),x\in \Omega ,t>\tau ,$$

$$u{|}_{\partial \Omega }=0,u(x,\tau )=u_0^{\tau },u_t(x,\tau )=u_1^{\tau },\tau \in \mathbb{R},$$

$$L^p=L^p(\Omega ),H^k=W^{k,2}(\Omega ),H_0^k=W_0^{k,2}(\Omega ),\Vert \cdot {\Vert }_p=\Vert \cdot {\Vert }_{L^p},\Vert \cdot \Vert =\Vert \cdot {\Vert }_{L^2},$$

$$\mathcal{H}=(H_0^1\cap L^{p+1})\times L^2,{\mathcal{H}}_{-1}=H_0^1\times H^{-1},$$

$$\Vert (u,v){\Vert }_{\mathcal{H}}^2=\Vert u{\Vert }_{H^1}^2+\Vert u{\Vert }_{p+1}^2+\Vert v{\Vert }^2.$$

$$c_0|s{|}^{p-1}-c_1\le f^{\prime }(s)\le c_2(1+|s{|}^{p-1}),\forall s\in \mathbb{R},$$

$$L_b^2(\mathbb{R};L^2)=\{\varphi \in L_{loc}^2(\mathbb{R};L^2)|\Vert \varphi {\Vert }_{L_b^2(\mathbb{R};L^2)}^2=\underset{t\in \mathbb{R}}{\sup }{\int }_t^{t+1}\Vert \varphi (s){\Vert }^{2}ds<+\infty \}.$$

$$u_{tt}-\sigma (\Vert \nabla u{\Vert }^2)\Delta u_t-\varphi (\Vert \nabla u{\Vert }^2)\Delta u+f(u)=h(x).$$

$$(u^n,v^n)\to (u,v)in{\mathcal{H}}_{ps}if\; and\; only\; if$$

$$(u^n,v^n)\to (u,v)in{H}_0^1\times L^{2}and{u}^n\rightharpoonup uin{L}^{p+1},$$

$$u_{tt}-\alpha \Delta u_t-(\beta +\gamma \Vert \nabla u{\Vert }^{\frac{\rho }{2}})\Delta u+h(u_t)+f(u,t)=g(x,t)$$

$$f^{\prime }(u)\ge -l,|f^{\prime }(u)|\le C(1+|u{|}^{p^{\ast }-1}),$$

$$\underset{|u|\to \infty }{\mathrm{liminf}}\frac{uf(u)-kF(u)}{u^2}\ge 0,\underset{|u|\to \infty }{\mathrm{liminf}}\frac{F(u)}{u^2}\ge 0,$$

$$U_{\sigma }(t,s)U_{\sigma }(s,\tau )=U_{\sigma }(t,\tau ),\forall t\ge s\ge \tau ,\tau \in \mathbb{R},$$

$$U_{\sigma }(\tau ,\tau )=I(identity\; operator),\tau \in \mathbb{R}.$$

$$U_{\sigma }(t+s,\tau +s)=U_{T(s)\sigma }(t,\tau ),\forall \sigma \in \Sigma ,t\ge \tau ,\tau \in \mathbb{R},s\ge 0.$$

$$\bigcup _{\sigma \in \Sigma }{U}_{\sigma }(t,\tau )B\subset B_0,\forall t\ge T_0.$$

$$\underset{s\to +\infty }{\lim }{\mathrm{dist}}_E\{U(t,t-s)D,\mathcal{A}(t)\}=0.$$

$${\mathrm{dist}}_E\{A,B\}=\underset{x\in A}{\sup }\underset{y\in B}{\inf }\Vert x-y{\Vert }_E,A,B\subset E.$$

$$\underset{t\to \infty }{\lim }\underset{\sigma \in \Sigma }{\sup }{\mathrm{dist}}_E\{U_{\sigma }(t,\tau )B,{\mathcal{A}}_{\Sigma }\}=0;$$

$${\mathcal{K}}_{\sigma }=\{u(\cdot )|U_{\sigma }(t,\tau )u(\tau )=u(t),\Vert u(t){\Vert }_E\le C_u,\forall t\ge \tau ,\tau \in \mathbb{R}\}$$

$${\mathcal{A}}_{\Sigma }={\omega }_{0,\Sigma }(B_0)=\bigcup _{\sigma \in \Sigma }{\mathcal{K}}_{\sigma }(s),\forall s\in \mathbb{R},$$

$$\omega_{0,\Sigma}(B_{0})=\bigcap_{t\geq 0}\Big{[}\bigcup_{\sigma\in\Sigma}\bigcup_{s\geq t}U_{\sigma}(s,0)B_{0}\Big{]}_{E},$$

$$\underset{k\to \infty }{\lim }\underset{l\to \infty }{\lim }\varphi (x_{n_k},x_{n_l};{\sigma }_{n_k},{\sigma }_{n_l})=0.$$

$$\Vert U_{{\sigma }_1}(T,0)x-U_{{\sigma }_2}(T,0)y{\Vert }_E\le \delta +{\varphi }_T(x,y;{\sigma }_1,{\sigma }_2),\forall x,y\in B_0,{\sigma }_1,{\sigma }_2\in \Sigma .$$

$$U_{\sigma }(t,\tau )=U_{{\sigma }^{\prime }}(t-\tau +{\tau }_0,{\tau }_0),\forall t\ge \tau ,{\tau }_0\in \mathbb{R}.$$

$$W=\{u\in L^p(0,T;X)|u_t\in L^1(0,T;Y)\},with1\le p<\infty ,$$

$$W_1=\{u\in L^{\infty }(0,T;X)|u_t\in L^r(0,T;Y)\},withr>1.$$

$$W\hookrightarrow \hookrightarrow L^p(0,T;B),W_1\hookrightarrow \hookrightarrow C([0,T];B).$$

$$T(h):L_{loc}^2(\mathbb{R};L^2)\to L_{loc}^2(\mathbb{R};L^2),T(h)g(s)=g(s+h),s,h\in \mathbb{R}.$$

$$\Sigma_{0}=\{T(h)g_{0}|h\in\mathbb{R}\},\ \ \Sigma=\mathcal{H}(g_{0})=\big{[}\Sigma_{0}\big{]}_{L^{2,w}_{loc}(\mathbb{R};L^{2})},$$

$$u^n\to uin\Sigma if\; and\; only\; if{u}^n\rightharpoonup uin{L}^2(t_1,t_2;L^2),\forall [t_1,t_2]\subset \mathbb{R}.$$

$$\underset{g\in \Sigma }{\sup }\Vert g{\Vert }_{L_b^2(\mathbb{R};L^2)}\le \Vert g_0{\Vert }_{L_b^2(\mathbb{R};L^2)},\underset{g\in \Sigma }{\sup }\Vert {\partial }_{t}g{\Vert }_{L_b^2(\mathbb{R};L^2)}\le \Vert {\partial }_t{g}_0{\Vert }_{L_b^2(\mathbb{R};L^2)},$$

Full text

Keywords: Non-autonomous Kirchhoff wave models; perturbed parameter; supercritical nonlinearity; uniform attractor; pullback attractor; upper semicontinuity.

1 Introduction

In this paper, we are concerned with the existence and upper semicontinuity of uniform attractors of the perturbed non-autonomous Kirchhoff wave equations with strong damping and supercritical nonlinearity:

[TABLE]

where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ ($N\geq 3$) with the smooth boundary $\partial\Omega,\epsilon\in[0,1]$ is a perturbed parameter. Throughout this paper we use the following notations:

[TABLE]

with $p\geq 1.$ The sign $H_{1}\hookrightarrow H_{2}$ denotes that the space $H_{1}$ continuously embeds into $H_{2}$ and $H_{1}\hookrightarrow\hookrightarrow H_{2}$ denotes that $H_{1}$ compactly embeds into $H_{2}$. We denote the phase spaces

[TABLE]

which are equipped with usual graph norms. For example,

[TABLE]

Assumption 1.1**.**

(i) $f\in C^{1}(\mathbb{R})$ and

[TABLE]

with some $\frac{N+2}{N-2}=p^{*}<p<p^{**}=\frac{N+4}{(N-4)^{+}}$, where $c_{i}$ are positive constants and $a^{+}=\max\{a,0\}$;

(ii) $(u^{\tau}_{0},u^{\tau}_{1})\in\mathcal{H}$ with $\|(u^{\tau}_{0},u^{\tau}_{1})\|_{\mathcal{H}}\leq R,g,\partial_{t}g\in L^{2}_{b}(\mathbb{R};L^{2})$, where

[TABLE]

When $N=1$, Eq. (1.1), without strong damping $-\Delta u_{t}$, was introduced by Kirchhoff [11] to describe the nonlinear vibrations of an elastic stretched string. In real process, dissipation plays an important spreading role for the energy gather arising from the nonlinearity. So the researches on the Kirchhoff wave equations with different type of dissipations have attracted considerable attention, the well-posedness and asymptotic behavior of solutions to the Kirchhoff wave models with dissipation $-\Delta u_{t}$ or $u_{t}$ or $h(u_{t})$ (with $h(s)s\geq 0$) have been well investigated by many authors (see [1, 3, 18, 20, 22, 23, 24] and references therein).

Recently, Chueshov [6] studied the well-posedness and longtime dynamics for the autonomous Kirchhoff wave model with strong nonlinear damping

[TABLE]

A major breakthrough is that he finds a supercritical exponent $p^{**}\equiv\frac{N+4}{(N-4)^{+}}$ and showes that when the growth exponent $p$ of the nonlinearity $f(u)$ is up to the supercritical range: $1\leq p<p^{**}$, the IBVP of Eq. (1.4) is still well-posed and the related solution semigroup has a partially strong global attractor $\mathcal{A}_{ps}$, i.e., the compactness and attractiveness of $\mathcal{A}_{ps}$ are in the phase space $\mathcal{H}_{ps}=(H^{1}_{0}\cap L^{p+1,w})\times L^{2}$, which is equipped with the partially strong topology:

[TABLE]

where the sign $``\rightharpoonup"$ denotes weak convergence. In particular, in the non-supercritical case: $1\leq p\leq p^{*}\equiv\frac{N+2}{N-2}$, the partially strong topology becomes the strong one. By the way, here the growth exponent $p^{*}$ is said to be critical relative to the natural energy space $\mathcal{H}=(H_{0}^{1}\cap L^{p+1})\times L^{2}$ for $H^{1}\hookrightarrow L^{p+1}$ as $p\leq p^{*}$, but the Sobolev embedding ceases to be effective as $p>p^{*}$. For the related researches on this topic, one can see also [7, 10, 15]. Recently, Ding, Yang and Li [7] removed the restriction of partially strong topology in [6].

Uniform attractor and pullback attractor (see Def. 2.2 and Def. 2.3 below) are two basic concepts to study the longtime dynamics of non-autonomous evolution equations with various dissipations (cf. [4, 9, 26, 28]). Although there have been some researches on the global attractors of autonomous Kirchhoff wave equations with strong damping (cf. [6, 10, 15, 20, 21, 30, 31, 32]), there are only a few recent results on the longtime dynamics of more complicated non-autonomous ones ([8, 29]). We refer to [8] for the investigations on the existence of the kernel $\mathcal{K}$ and the Hausdorff dimension of the kernel sections $\mathcal{K}(s)$ for strongly damped non-autonomous Kirchhoff wave models

[TABLE]

in a bounded domain $\Omega\subset\mathbb{R}^{N}(N=1,2,3)$ with Dirichlet boundary condition, where $\alpha>0,\beta>0,\rho\geq-1,\gamma\geq 0$ and the source term $f(u,t)$ is of subcritical growth on $u$.

Recently, Wang and Zhong [29] studied the existence and the upper semi-continuity of pullback attractors of problem (1.1)-(1.2). Under the critical nonlinearity assumptions:

[TABLE]

where $F(u)=\int_{0}^{u}f(s)ds$, they established the existence of pullback attractors and their upper semicontinuity on the perturbed parameter $\epsilon$.

But there are still some unsolved questions. For example, for the perturbed non-autonomous Kirchhoff wave model (1.1), if the nonlinearity $f(u)$ is of the supercritical growth $p:p^{*}\leq p<p^{**}$, what about the existence and structure of its uniform attractor and pullback attractor? What about the stability of the attractors on the perturbed parameter $\epsilon$?

The purpose of the present paper is to solve these questions. It proves that in supercritical nonlinearity case $p^{*}<p<p^{**}$:

(i) the related family of processes $\{U^{\epsilon}_{g}(t,\tau)\},g\in\Sigma$ has in $\mathcal{H}$ a compact uniform attractor $\mathcal{A}^{\epsilon}_{\Sigma}$ for each $\epsilon\in[0,1]$ and its structure is shown (see Theorem 4.3);

(ii) the family of compact uniform attractor $\mathcal{A}^{\epsilon}_{\Sigma}$ is upper semicontinuous on the perturbed parameter $\epsilon$ in the sense of $\mathcal{H}_{ps}$ topology (i.e., partially strong topology) (see Corollary 5.4).

As a consequence, for any fixed $g\in\Sigma$ (the symbol space), the family of all kernel sections $\mathcal{A}^{\epsilon}_{g}=\{\mathcal{K}^{\epsilon}_{g}(t)\}_{t\in\mathbb{R}}$ is the pullback attractor of the process $\{U^{\epsilon}_{g}(t,\tau)\}$ in $\mathcal{H}$ for each $\epsilon\in[0,1]$ (cf. [4]), and it is also upper semicontinuous on $\epsilon$ in the sense of $\mathcal{H}_{ps}$ topology (see Corollary 5.4).

In particular, for autonomous case, i.e., $g(x,t)\equiv g(x)$, the related process $\{U^{\epsilon}_{g}(t,\tau)\}$ becomes the solution semigroup $S^{\epsilon}(t)$ acting on the phase space $\mathcal{H}$ for each $\epsilon\in[0,1]$, and the related pullback attractor becomes the global attractor $\mathcal{A}^{\epsilon}$ of $S^{\epsilon}(t)$ in $\mathcal{H}$, which is upper semicontinuous on $\epsilon$ in the sense of $\mathcal{H}_{ps}$ topology.

The main contributions of the paper are that under the assumptions that the external force $g$ is translation bounded (rather than translation compact as usual), and the the nonlinearity $f(u)$ is of supercritical growth $p:p^{*}<p<p^{**}$, by combining newly developed criterion of compensated compactness [27], quasi-stabilizability estimates method [5] and J. Ball’s technique [2], we prove the existence of the uniform attractor $A^{\epsilon}_{\Sigma}$ of problem (1.1)-(1.2) and show their upper semicontinuity on the perturbed parameter $\epsilon$ in the sense of partially strong topology. These results not only extend Chueshov’s work on autonomous Kirchhoff models in [6] to non-autonomous ones but also extend Wang and Zhong’s results on pullback attractor [29] to the supercritical nonlinearity case.

Recently, many authors devote to study the uniform attractor of non-autonomous dissipative PDEs with non translation compact external forces. They introduce several new classes of external forces that are not translation compact, but nevertheless allow the attraction in a strong topology of the phase space and give some criteria on this kind of uniform attractor and applications of them (cf. [12, 13, 14, 16, 17, 19, 27, 33, 34]).

We show in the present paper that the weak solutions of non-autonomous Kirchhoff wave model (1.1)-(1.2) are of higher partial regularity when $t>\tau$, which results in that not only the requirement for the external force $g:g,\partial_{t}g\in L^{2}_{b}(\mathbb{R};L^{2})$ is natural but also permits non translation compact external forces $g$.

The paper is organized as follows. In Section 2, we introduce some preliminaries. In Section 3, we give some results on the well-posedness. In Section 4, we discuss the existence of uniform attractors. In Section 5, we investigate the upper semicontinuity of the uniform attractors on the perturbed parameter $\epsilon$.

2 Preliminaries

Definition 2.1**.**

(i) The family of sets $\{U_{\sigma}(t,\tau)|t\geq\tau,\tau\in\mathbb{R}\},\sigma\in\Sigma$ (parameter set) is said to be a family of processes acting on Banach space $E$ if for each $\sigma\in\Sigma$, $\{U_{\sigma}(t,\tau)|t\geq\tau,\tau\in\mathbb{R}\}$ is a process acting on $E$, i.e., the two-parameter mappings from $E$ to $E$ satisfying

[TABLE]

And the set $\Sigma$ is said to be the symbol space and $\sigma\in\Sigma$ to be a symbol.

(ii) Let $\{T(t)\}_{t\geq 0}$ be a translation semigroup acting on $\Sigma$. The family of processes $\{U_{\sigma}(t,\tau)\},\sigma\in\Sigma$ is said to be satisfy the translation identity if

[TABLE]

(iii) A bounded subset $B_{0}\subset E$ is said to be a bounded uniformly ($w.r.t.\ \sigma\in\Sigma$) absorbing set of the family of processes $\{U_{\sigma}(t,\tau)\},\sigma\in\Sigma$ if for any $\tau\in\mathbb{R}$ and bounded subset $B\subset X$ there exists a $T_{0}=T_{0}(B,\tau)\geq\tau$ such that

[TABLE]

Definition 2.2**.**

A family of nonempty compact subsets $\{\mathcal{A}(t)\}_{t\in\mathbb{R}}$ of $E$ is said to be a pullback attractor of the process $U(t,\tau)$ if it is invariant, i.e., $U(t,s)\mathcal{A}(s)=\mathcal{A}(t),\ t\geq s$, and it pullback attracts all the bounded subsets of $E$, i.e., for every bounded subset $D\subset E$ and $t\in\mathbb{R}$,

[TABLE]

Here, $\mathrm{dist}_{E}\{\cdot,\cdot\}$ is the Hausdorff semidistance in $E$, i.e.,

[TABLE]

Definition 2.3**.**

A closed set $\mathcal{A}_{\Sigma}\subset E$ is said to be the uniform ($w.r.t.\ \sigma\in\Sigma$) attractor of the family of processes $\{U_{\sigma}(t,\tau)\},\sigma\in\Sigma$ if

(i) (Attractiveness) $\mathcal{A}_{\Sigma}$ uniformly ($w.r.t.\ \sigma\in\Sigma$) attracts all the bounded subsets in $E$, i.e., for every bounded subset $B\subset X$ and $\tau\in\mathbb{R}$,

[TABLE]

(ii) (Minimality) for any closed set $\mathcal{A}^{\prime}\subset E$, if $\mathcal{A}^{\prime}$ is of property (i), then $\mathcal{A}_{\Sigma}\subset\mathcal{A}^{\prime}$.

Definition 2.4**.**

(i) For any fixed $\sigma\in\Sigma$, the set of all bounded full trajectories of the process $U_{\sigma}(t,\tau)$:

[TABLE]

is said to be the kernel of the process $U_{\sigma}(t,\tau)$. The set $\mathcal{K}_{\sigma}(s)=\{u(s)|u(\cdot)\in\mathcal{K}_{\sigma}\}$ is said to be the kernel section at time $t=s,s\in\mathbb{R}$.

(ii) The family of processes $\{U_{\sigma}(t,\tau)\},\sigma\in\Sigma$ is said to be uniformly ($w.r.t.\ \sigma\in\Sigma$) asymptotically compact on $E$, if for any $\tau\in\mathbb{R}$, bounded sequences $\{\xi_{n}\}\subset E,\{\sigma_{n}\}\subset\Sigma$ and sequence $\{t_{n}\}\subset\mathbb{R}$ with $t_{n}\geq\tau$ and $t_{n}\rightarrow+\infty$, the sequence $\{U_{\sigma_{n}}(t_{n},\tau)\xi_{n}\}$ is precompact in $E$ (cf. [19]).

(iii) The family of processes $\{U_{\sigma}(t,\tau)\},\sigma\in\Sigma$ is said to be norm-to-weak continuous, if for any fixed $t$ and $\tau\in\mathbb{R}$ with $t\geq\tau$, for any sequence $\{(x_{n},\sigma_{n})\}\subset E\times\Sigma,(x_{n},\sigma_{n})\rightarrow(x,\sigma)$ in $E\times\Sigma$ imply that $U_{\sigma_{n}}(t,\tau)x_{n}\rightharpoonup U_{\sigma}(t,\tau)x$ in $E$.

Lemma 2.5**.**

[27]** Assume that $\Sigma$ is a compact metric space, the translation semigroup $\{T(t)\}_{t\geq 0}$ is continuous in $\Sigma$, the family of processes $\{U_{\sigma}(t,\tau)\},\sigma\in\Sigma$ satisfies the translation identity (2.1) and

(i) it is norm-to-weak continuous;

(ii) it has a bounded uniformly ($w.r.t.\ \sigma\in\Sigma$) absorbing set $B_{0}$ in $E$;

(iii) it is uniformly ($w.r.t.\ \sigma\in\Sigma$) asymptotically compact in $E$.

Then it has a compact uniform ($w.r.t.\ \sigma\in\Sigma$) attractor $\mathcal{A}_{\Sigma}$, and

[TABLE]

where $\mathcal{K}_{\sigma}$ is the kernel of the process $U_{\sigma}(t,\tau),\omega_{0,\Sigma}(B_{0})$ is the uniform $\omega$-limit set of $B_{0}$ at $t=0$, i.e.,

[TABLE]

and the sign $[\,\cdot\,]_{E}$ denotes the closure in $E$.

Definition 2.6**.**

Let $\Sigma$ be a symbol space and $B$ be a bounded subset in Banach space $E$. A function $\phi(\cdot,\cdot;\cdot,\cdot)$ defined on $(B\times B)\times(\Sigma\times\Sigma)$ is said to be a contractive function if for any sequences $\{x_{n}\}\subset B$ and $\{\sigma_{n}\}\subset\Sigma$, there exist subsequences $\{x_{n_{k}}\}\subset\{x_{n}\}$ and $\{\sigma_{n_{k}}\}\subset\{\sigma_{n}\}$ such that

[TABLE]

Lemma 2.7**.**

[27]** Assume that the family of processes $\{U_{\sigma}(t,\tau)\},\sigma\in\Sigma$ satisfies translation identity (2.1), and the following conditions holds:

(i) it has a bounded uniformly ($w.r.t.\ \sigma\in\Sigma$) absorbing set $B_{0}\subset E$;

(ii) for any $\delta>0$ there exist $T=T(B_{0},\delta)>0$ and a contractive function $\phi_{T}$ defined on $(B_{0}\times B_{0})\times(\Sigma\times\Sigma)$ such that

[TABLE]

Then the family of processes $\{U_{\sigma}(t,\tau)\},\sigma\in\Sigma$ is uniformly ($w.r.t.\ \sigma\in\Sigma$) asymptotically compact on $E$.

Lemma 2.8**.**

[12]** Let the family of processes $\{U_{\sigma}(t,\tau)\},\sigma\in\Sigma$ satisfy the translation identity (2.1) and the symbol space $\Sigma$ be translation invariant, i.e., $T(h)\Sigma=\Sigma$ for all $h\geq 0$. Then for every $\tau\in\mathbb{R}$ and $\sigma\in\Sigma$, there exists at least one $\sigma^{\prime}\in\Sigma$ satisfying

[TABLE]

Lemma 2.9**.**

[25]** Let $X,B$ and $Y$ be Banach spaces, $X\hookrightarrow\hookrightarrow B\hookrightarrow Y,$

[TABLE]

Then,

[TABLE]

3 Well-posedness

In this section, we discuss the well-posedness of problem (1.1)-(1.2). We first define a symbol space generated by a fixed external force term $g_{0}$, with $g_{0},\partial_{t}g_{0}\in L^{2}_{b}(\mathbb{R};L^{2})$.

Define the translation operator

[TABLE]

Obviously, $\{T(h)\}_{h\in\mathbb{R}}$ constitutes a translation group on $L^{2}_{loc}(\mathbb{R};L^{2})$. Let

[TABLE]

and $\Sigma$ be equipped with $L^{2,w}_{loc}(\mathbb{R};L^{2})$ topology, i.e.,

[TABLE]

Then $\Sigma$ is a compact metric space,

[TABLE]

and $\{T(t)\}_{t\in\mathbb{R}}$ is continuous and invariant in $\Sigma$, i.e., $T(h)\Sigma=\Sigma,\ \forall h\in\mathbb{R}$ (cf. [4]).

Repeating the same arguments as in [6] (where the well-posedness of problem (1.1)-(1.2) has been established for the autonomous case: $g(x,t)\equiv g(x)$ ) except for the treatment of $g(x,t)$ one easily gets the following theorem.

Theorem 3.1**.**

Let Assumption 1.1 be valid, with $g\in\Sigma$. Then problem (1.1)-(1.2) admits a unique weak solution $u^{\epsilon}$, with $(u^{\epsilon},u^{\epsilon}_{t})\in C([\tau,T];\mathcal{H})$ for each $\epsilon\in[0,1]$, and

[TABLE]

where $K=C(\tau,T,R,\|g\|_{L^{2}_{b}(\mathbb{R};L^{2})})$ is a positive constant. Moreover, the solution is of the following properties:

(i) (Partial regularity when $t>\tau$)

[TABLE]

where $K_{1}=C(T-\tau,R,\|g_{0}\|_{L^{2}_{b}(\mathbb{R};L^{2})},\|\partial_{t}g_{0}\|_{L^{2}_{b}(\mathbb{R};L^{2})})$;

(ii) (Energy identity)

[TABLE]

where

[TABLE]

(iii) (Stability and quasi-stability in $\mathcal{H}_{-1}$) the following Lipschitz stability

[TABLE]

and quasi-stability

[TABLE]

hold for $z=u^{\epsilon,1}-u^{\epsilon,2}$, where $u^{\epsilon,1},u^{\epsilon,2}$ are two weak solutions of problem (1.1)-(1.2) corresponding to initial data $(u^{\epsilon,i}(\tau),u^{\epsilon,i}_{t}(\tau))\in\mathcal{H}$, with $\|(u^{\epsilon,i}(\tau),u^{\epsilon,i}_{t}(\tau))\|_{\mathcal{H}}\leq R$, and $g_{i}\in L^{2}_{b}(\mathbb{R};L^{2})$, respectively.

For any $g\in\Sigma$, we define the solution operator

[TABLE]

where $u^{\epsilon}$ is a weak solution of problem (1.1)-(1.2). Theorem 3.1 shows that $\{U^{\epsilon}_{g}(t,\tau)\},g\in\Sigma,\epsilon\in[0,1]$ is a family of processes acting on the phase space $\mathcal{H}$. The uniqueness of weak solutions implies the translation identity

[TABLE]

4 Existence of uniform attractors

For simplicity, we omit the superscript $\epsilon$ and denote $u=u^{\epsilon}$ in the following.

Lemma 4.1**.**

Let Assumption 1.1 be valid, with $g\in\Sigma$. Then

1. $(i)$

For any sequence $\{(\xi_{n},g_{n})\}\subset\mathcal{H}\times\Sigma$ with $(\xi_{n},g_{n})\rightarrow(\xi,g)$ in $\mathcal{H}_{-1}\times\Sigma$, we have

[TABLE] 2. $(ii)$

The family of processes $\{U^{\epsilon}_{g}(t,\tau)\},g\in\Sigma$ is norm-to-weak continuous for each $\epsilon\in[0,1]$.

Proof.

(i) The fact $g_{n}\rightarrow g$ in $\Sigma$ implies that

[TABLE]

Indeed, it follows from estimate (3.2) that both the sequences $\{g_{n}\}$ and $\{\partial_{t}g_{n}\}$ are bounded in $L^{2}(\tau,t;L^{2})$, which implies that $\{g_{n}\}$ is precompact in $L^{2}(\tau,t;H^{-1})$ for $L^{2}\hookrightarrow\hookrightarrow H^{-1}$ (see Lemma 2.9). So formula (4.2) holds. The combination of (4.2) and stability estimate (3.6) yields (4.1).

(ii) Let $(\xi_{n},g_{n})\rightarrow(\xi,g)$ in $\mathcal{H}\times\Sigma$. By (4.1),

[TABLE]

By the boundedness of $\{(u^{n},u_{t}^{n})(t)\}$ in $\mathcal{H}$ (see (3.3)),

[TABLE]

Therefore,

[TABLE]

∎

Lemma 4.2**.**

Let Assumption 1.1 be valid, with $g\in\Sigma$. Then the family of processes $\{U^{\epsilon}_{g}(t,\tau)\},g\in\Sigma,\epsilon\in[0,1]$ has a uniformly ($w.r.t.\ g\in\Sigma$ and $\epsilon\in[0,1]$) absorbing set $\mathcal{B}=\{\xi\in\mathcal{H}|\|\xi\|_{\mathcal{H}}\leq R_{0}\}$.

Proof.

Using the multiplier $u_{t}+\delta u\ (=u^{\epsilon}_{t}+\delta u^{\epsilon}$) in Eq. (1.1), we obtain

[TABLE]

where $\xi_{u}=(u,u_{t})$,

[TABLE]

Assumption (1.3) implies that

[TABLE]

Thus a simple calculation shows that

[TABLE]

for $\delta>0$ suitably small, where $\lambda_{1}$ is the first eigenvalue of $-\Delta$ with Dirichlet boundary condition and $\kappa$ is a small positive constant. Hence,

[TABLE]

for all $\epsilon\in[0,1],g\in\Sigma$ and $\xi_{u}(\tau)\in\mathcal{H}$, where $\xi_{u}(t)=U^{\epsilon}_{g}(t,\tau)\xi_{u}(\tau),Q$ is a monotone positive function.

Let

[TABLE]

Estimate (4.6) shows that $\mathcal{B}$ is a uniformly ($w.r.t.\ g\in\Sigma$ and $\epsilon\in[0,1]$) absorbing set of the family of processes $\{U^{\epsilon}_{g}(t,\tau)\},g\in\Sigma,\epsilon\in[0,1]$. ∎

Theorem 4.3**.**

Let Assumption 1.1 be valid, with $g\in\Sigma$. Then the family of processes $\{U^{\epsilon}_{g}(t,\tau)\},g\in\Sigma$ has in $\mathcal{H}$ a compact uniform ($w.r.t.\ g\in\Sigma$) attractor $\mathcal{A}_{\Sigma}^{\epsilon}$ for each $\epsilon\in[0,1]$, and

[TABLE]

Proof.

Since the family of processes $\{U^{\epsilon}_{g}(t,\tau)\},g\in\Sigma$ satisfies translation identity (3.8), it is norm-to-weak continuous for each $\epsilon\in[0,1]$ (see Lemma 4.1) and has a uniformly ($w.r.t.\ g\in\Sigma$ and $\epsilon\in[0,1]$) absorbing set $\mathcal{B}$ (see Lemma 4.2), by Lemma 2.5, it is sufficient to prove Theorem 4.3 to show the precompactness of the sequence $\{U^{\epsilon}_{g_{n}}(t_{n},\tau)\xi_{n}\}$ in $\mathcal{H}$, where $t_{n}\rightarrow+\infty$ as $n\rightarrow\infty$ (see Def. 2.4: (ii)). By translation identity (3.8),

[TABLE]

Without loss of generality, it is enough to show that for every $\epsilon\in[0,1]$, any sequences $\{g_{n}\}\subset\Sigma$, $\{\xi_{n}\}\subset\mathcal{B}$ and $\tau_{n}\rightarrow-\infty$, the sequence $\{U^{\epsilon}_{g_{n}}(0,\tau_{n})\xi_{n}\}$ is precompact in $\mathcal{H}$.

Let

[TABLE]

Due to Lemma 2.8 (taking $\tau_{0}=0$ there) and the fact that $\mathcal{B}$ is a uniformly ($w.r.t.\ g\in\Sigma$) absorbing set of the family of processes $\{U^{\epsilon}_{g}(t,\tau)\},g\in\Sigma$, there exists a positive constant $T_{0}$ independent of $\tau$ such that

[TABLE]

For any fixed $T\in\mathbb{N}$, there exists a $N>0$ such that $-T\geq\tau_{n}+T_{0}$ as $n\geq N$. Hence when $n\geq N$, by (4.10),

[TABLE]

Therefore (see (3.3)),

[TABLE]

and (subsequence if necessary)

[TABLE]

where we have used the compactness of $\Sigma$. By Lemma 2.9,

[TABLE]

where $\delta\in(0,1)$. It follows from estimate (3.7) that

[TABLE]

for any $\epsilon\in[0,1],x,y\in\mathcal{B}$ and $g_{1},g_{2}\in\Sigma$, where $C=C(t,R_{0},\|g_{0}\|_{L^{2}_{b}(\mathbb{R};L^{2})})$. For any sequence $\{g_{n}\}\subset\Sigma,\{g_{n}\}$ is precompact in $L^{2}(0,t;H^{-1})$ for $g_{n}\rightarrow g$ in $\Sigma$ (see (4.2)). By the similar arguments as (4), we obtain that

[TABLE]

Thus, it follows from (4.13) that for any $\delta>0$, there exist $T=T(\mathcal{B},\delta)>0$ and a contractive function

[TABLE]

defined on $(\mathcal{B}\times\mathcal{B})\times(\Sigma\times\Sigma)$ such that

[TABLE]

By Lemma 2.7, the family of processes $\{U^{\epsilon}_{g}(t,\tau)\},g\in\Sigma$ is uniformly ($w.r.t.\ g\in\Sigma$) asymptotically compact in $\mathcal{H}_{-1}$. Therefore (subsequence if necessary),

[TABLE]

By formula (4.1) and the uniqueness of the limit,

[TABLE]

So

[TABLE]

By the standard diagonal process, we can extract a subsequence (still denoted by itself) such that (4.11) and (4.14)-(4.16) hold for all $T\in\mathbb{N}$.

Rewrite energy identity (3.5) as the form

[TABLE]

Using the multiplier $\delta u$ in Eq. (1.1) and adding the resulting expression to (4.17), we obtain

[TABLE]

where $\Gamma(u,u_{t})$ is as shown in (4.4) and

[TABLE]

It follows from (4.18) that

[TABLE]

and the formula (4.19) also holds for $(u,u_{t})$. By virtue of (4.11)-(4), (4.16) and the Lebesgue dominated convergence theorem,

[TABLE]

It follows from (4.11) that

[TABLE]

By (4),

[TABLE]

hence by formula (4.5) and the Fatou lemma,

[TABLE]

The combination of (4.20)-(4.22) yields

[TABLE]

Therefore, taking account of the boundedness of $\Gamma((u^{n},u^{n}_{t})(-T))$, we infer from (4.19) and (4.23) that

[TABLE]

Letting $T\rightarrow+\infty$, we obtain

[TABLE]

where we have used (4.10)-(4.11), (4.15) and the Fatou lemma in the second inequality. Therefore,

[TABLE]

which implies (see (4.4))

[TABLE]

By (4.5), the Fatou lemma and (4.24),

[TABLE]

where $C_{1}=\frac{c_{0}}{2p(p+1)}$, that is,

[TABLE]

By (4.11),

[TABLE]

The combination of (4.24)-(4.26) and the uniform convexity of $L^{p+1}$ and $L^{2}$ yields

[TABLE]

i.e., the family of processes $\{U^{\epsilon}_{g}(t,\tau)\},g\in\Sigma$ is uniformly ($w.r.t.\ g\in\Sigma$) asymptotically compact in $\mathcal{H}$. Therefore, by Lemma 2.5, we get the conclusion of Theorem 4.3. ∎

5 Upper semicontinuity of the uniform attractors

In this section, we discuss the upper semicontinuity ($w.r.t.\ \epsilon$) of the uniform attractors $\mathcal{A}_{\Sigma}^{\epsilon}$.

Theorem 5.1**.**

Let Assumption 1.1 be valid, with $g\in\Sigma$. Then the uniform attractors $\mathcal{A}_{\Sigma}^{\epsilon}$ as shown in Theorem 4.3 is upper semicontinuous at the point $\epsilon_{0}\geq 0$ in the sense of $\mathcal{H}_{-1}$ topology, i.e.,

[TABLE]

and so does the kernel section $\mathcal{K}_{g}^{\epsilon}(s)$, i.e.,

[TABLE]

In order to prove Theorem 5.1, we first give following lemmas.

Lemma 5.2**.**

(Lipschitz stability) Under the assumptions of Theorem 5.1, we have

[TABLE]

for any $\xi_{i}\in\mathcal{B}$ and $\epsilon_{i}\in[0,1]\ (i=1,2)$, where $C_{K}=C(\tau,T,R_{0},\|g_{0}\|_{L^{2}_{b}(\mathbb{R};L^{2})})$.

Proof.

Let $(u^{i}(t),u_{t}^{i}(t))=U^{\epsilon_{i}}_{g}(t,\tau)\xi_{i}$. It follows from estimate (3.3) that

[TABLE]

where $K=K(\tau,T,R_{0},\|g_{0}\|_{L^{2}_{b}(\mathbb{R};L^{2})})>0$. Then $z=u^{1}-u^{2}$ solves

[TABLE]

Using the multiplier $(-\Delta)^{-1}z_{t}+\delta z$ in Eq. (5.5), we obtain

[TABLE]

where $\xi_{z}=(z,z_{t})$,

[TABLE]

for $\delta>0$ suitably small. Obviously,

[TABLE]

where we have used estimate (5.4). By Assumption (1.3), the Sobolev embedding $H^{2-\theta}\hookrightarrow L^{p+1}$ for $0<\theta\ll 1$ and the interpolation, we have

[TABLE]

Inserting above estimates into (5.6) yields

[TABLE]

Applying the Gronwall inequality to (5.7) over $(\tau,t)$ gives (5.3). ∎

Lemma 5.3**.**

Under the assumptions of Theorem 5.1, the family of processes $\{U^{\epsilon}_{g}(t,\tau)\},g\in\Sigma,\epsilon\in[0,1]$ has a uniformly ($w.r.t.\ g\in\Sigma$ and $\epsilon\in[0,1]$) absorbing set $\mathcal{B}_{0}$, which is bounded in $(H_{0}^{1}\cap L^{p+1})\times H_{0}^{1}$.

Proof.

By Lemma 4.2, there exists a $T>0$ such that

[TABLE]

Let

[TABLE]

Then $\mathcal{B}_{0}$ is the desired absorbing set. Indeed, for any bounded set $D\subset\mathcal{H}$, there exists a $t_{D}\geq 0$ such that

[TABLE]

When $t\geq t_{D}+T+1+\tau$, by Lemma 2.8, there exist at least one $g^{\prime}\in\Sigma$ such that

[TABLE]

for any $\epsilon\in[0,1],g\in\Sigma,\tau\in\mathbb{R}$, where we have used translation identity (3.8). Due to

[TABLE]

we infer from estimate (3.4) that $\mathcal{B}_{0}$ is bounded in $(H_{0}^{1}\cap L^{p+1})\times H_{0}^{1}$. ∎

Proof of Theorem 5.1.

If the formula (5.1) does not hold. There must exist $\delta>0,\epsilon_{0}\in[0,1],\{\epsilon_{n}\}\subset[0,1]$ with $\epsilon_{n}\rightarrow\epsilon_{0}$, and $\xi_{n}\in\mathcal{A}^{\epsilon_{n}}_{\Sigma}$ such that

[TABLE]

Due to $\mathcal{H}\hookrightarrow\mathcal{H}_{-1}$, we have

[TABLE]

which implies that there exists a $T>0$ such that when $t\geq T$,

[TABLE]

Due to $\xi_{n}\in\mathcal{A}_{\Sigma}^{\epsilon_{n}}=\omega^{\epsilon_{n}}_{0,\Sigma}(\mathcal{B})$, there exist $g_{n}\in\Sigma$, $\eta_{n}\in\mathcal{B}$ and $t_{n}\geq 2T$ such that

[TABLE]

Since

[TABLE]

where $z_{n}=U^{\epsilon_{n}}_{g_{n}}(t_{n}-T,0)\eta_{n}\in\mathcal{B}$ for $t_{n}-T\geq T$, we infer from Lemma 5.2 that there exists a $N>0$ such that

[TABLE]

for $\epsilon_{n}\rightarrow\epsilon_{0}$. Therefore, it follows from estimates (5.10)-(5.12) that

[TABLE]

which violates (5.9). Therefore, formula (5.1) holds.

Now, we give the proof of formula (5.2) by contradiction. If formula (5.2) does not hold, there must exist $s_{0}\in\mathbb{R},g\in\Sigma,\delta>0,\epsilon_{0}\in[0,1]$, sequences $\{\epsilon_{n}\}\subset[0,1]$ with $\epsilon_{n}\rightarrow\epsilon_{0}$ and $\xi_{n}\in\mathcal{K}^{\epsilon_{n}}_{g}(s_{0})$ such that

[TABLE]

On the other hand, the process $U^{\epsilon_{n}}_{g}(t,\tau)$ has a bounded full trajectory $\gamma_{n}=\{\xi_{u}^{n}(t)|t\in\mathbb{R}\}$ for each $n$ such that

[TABLE]

Formula (4.7) shows that $\xi_{u}^{n}(s)\in\mathcal{K}^{\epsilon_{n}}_{g}(s)\subset\mathcal{A}_{\Sigma}^{\epsilon_{n}},\ \forall s\in\mathbb{R}$. By formula (5.1) and the compactness of $\mathcal{A}_{\Sigma}^{\epsilon_{0}}$ in $\mathcal{H}_{-1}$, there must exist a $\xi_{u}(s)\in\mathcal{A}_{\Sigma}^{\epsilon_{0}}$ such that (subsequence if necessary),

[TABLE]

Then we infer form Lemma 5.2 that

[TABLE]

By the uniqueness of the limit,

[TABLE]

which means $\gamma=\{\xi_{u}(t)|t\in\mathbb{R}\}\in\mathcal{K}^{\epsilon_{0}}_{g}$ and $\xi_{u}(s_{0})\in\mathcal{K}^{\epsilon_{0}}_{g}(s_{0})$. Hence,

[TABLE]

which violates (5.13). Therefore, formula (5.2) holds. ∎

We consider the bounded uniformly absorbing set $\mathcal{B}_{0}$ as a topology space equipped with the partially strong topology as shown in (1.5). Since $\mathcal{B}_{0}$ is bounded in $(H_{0}^{1}\cap L^{p+1})\times H_{0}^{1}$, this topology can be defined by the following metric $\rho$:

[TABLE]

where $\xi_{u}=(u_{0},u_{1}),\xi_{v}=(v_{0},v_{1})\in\mathcal{B}_{0}$, $\{g_{n}\}\subset H^{-1}\cap L^{1+\frac{1}{p}}$ such that $\|g_{n}\|_{H^{-1}}=1$ and $span\{g_{n}|n\in\mathbb{N}\}$ is dense in $L^{1+\frac{1}{p}}$ (cf. [6]).

Corollary 5.4**.**

Let Assumption 1.1 be valid, with $g\in\Sigma$. Then

1. $(i)$

the compact uniform attractors $\mathcal{A}_{\Sigma}^{\epsilon}$ as shown in Theorem 4.3 is upper semicontinuous at the point $\epsilon_{0}\in[0,1]$ in the sense of partially strong topology, i.e.,

[TABLE]

where

[TABLE] 2. $(ii)$

for any fixed $g\in\Sigma$ and $\epsilon\in\Sigma$, the family of all kernel sections $\mathcal{A}^{\epsilon}_{g}=\{\mathcal{K}_{g}^{\epsilon}(t)\}_{t\in\mathbb{R}}$ is the pullback attractor of the process $\{U^{\epsilon}_{g}(t,\tau)\}$, and it is upper semicontinuous at the point $\epsilon_{0}\in[0,1]$ in the sense of partially strong topology, i.e.,

[TABLE]

Proof.

Since $\mathcal{A}_{\Sigma}^{\epsilon}$ is the compact uniform attractor of the family of processes $\{U^{\epsilon}_{g}(t,\tau\},g\in\Sigma$ and (4.7) holds, by the standard theory on the uniform attractor (cf. Chapter IV in [4]), for any fixed $g\in\Sigma$ and $\epsilon\in\Sigma$, the family of all kernel sections $\mathcal{A}^{\epsilon}_{g}=\{\mathcal{K}_{g}^{\epsilon}(t)\}_{t\in\mathbb{R}}$ is just a pullback attractor of the process $\{U^{\epsilon}_{g}(t,\tau\}$.

Due to

[TABLE]

we see from (5.17) that

[TABLE]

For any $\xi_{u}=(u_{0},u_{1}),\xi_{v}=(v_{0},v_{1})\in\mathcal{B}_{0}$, by the interpolation,

[TABLE]

Taking account of $\mathcal{A}^{\epsilon}_{\Sigma}\subset\mathcal{B}_{0}$ for all $\epsilon\in[0,1]$, we infer from (5.18)-(5.19) and Theorem 5.1 that

[TABLE]

∎