Existence of a Global Attractor for the Reaction–Diffusion Equation with Memory and Lower Regularity Terms

Abstract

This paper investigates the large time behavior for the reaction–diffusion equation with memory and the forcing term $g\in H^{-1}\left(\Omega \right).$ We prove the existence of a global attractor in $L^2\left(\Omega \right)\times L_{\mu }^2(\mathbb{R};H_0^1\left(\Omega \right))$. Due to the lower regularity of g, one can hardly use the traditional energy estimates to derive the existence of a bounded absorbing set in the higher regularity space and then the compactness of the semigroup. Here, we utilize the contractive function method to establish the asymptotic smoothness of the semigroup.

1. Introduction

We investigate the existence of a global attractor for the following reaction–diffusion equation with memory:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{cc}{u}_t-\mathsf{\Delta }u-{\int }_0^{\infty }k\left(s\right)\mathsf{\Delta }u(t-s)ds+f\left(u\right)=g,\hfill & \mathrm{in}\Omega ,t>0,\hfill \\ u\left(t\right)=u_0\left(t\right),\hfill & \mathrm{in}\Omega ,t⩽0,\hfill \\ {u|}_{\partial \Omega }=0,\hfill & t\in \mathbb{R},\hfill \end{array}\right.\end{array}\end{array}$$

(1)

in a bounded domain $\Omega \subset {\mathbb{R}}^n(n\ge 3)$, where $g\in H^{-1}\left(\Omega \right)$, and memory kernel function $k\left(s\right)$ is assumed to satisfy the posterior adaptation hypothesis, which will be detailed in the subsequent sections.

It is well established that numerous physical phenomena can be described more accurately by incorporating terms into the governing equations that account for the system’s historical behavior. In particular, models pertaining to high-viscosity fluids at low temperatures or the thermomechanical behavior of polymers (refer to [1,2] and the associated references) demonstrate the significant influence of historical effects. In certain mathematical formulations of physical equations, the presence of the memory term is particularly salient, as it is reasonable to assert that the evolution of a system is contingent upon its entire historical trajectory rather than solely its present state.

The long-time behavior of solutions of Equation (1) with memory has been studied extensively. In order to generate a dynamical system in an appropriate phase space, Dafermos [3] introduced an additional variable ${\eta }^t$, the (integrated) past history of u,

$$\begin{array}{c}\hfill {\eta }^t={\eta }^t(x,s)={\int }_0^{s}u(x,t-\tau )d\tau ,\end{array}$$

of which the evolution is ruled by a first-order hyperbolic equation. Thus, the original problem (1) can be transformed into a dynamical system on a phase space with two components

$$\left\{\begin{array}{c}{u}_t-\mathsf{\Delta }u-{\int }_0^{\infty }\mu \left(s\right)\mathsf{\Delta }{\eta }^t\left(s\right)ds+f\left(u\right)=g,\hfill \\ {\eta }_t^t=-{\eta }_s^t+u,\hfill \end{array}\right.$$

Following the Dafermos’ idea, the long-time behavior of solutions to the reaction–diffusion equations with memory have been studied largely. In [4], the existence of global attractors for reaction–diffusion systems with finite delay was obtained, while in [5], the authors studied the existence of trajectory attractors for reaction–diffusion equations with an infinite-delay memory term. In [6], the existence of global attractors in $L^2\left(\Omega \right)\times L_{\mu }^2(\mathbb{R};H_0^1\left(\Omega \right))$ was obtained for Equation (1) with critical nonlinearity. In [7], when the nonlinearity is subcritical, the existence of a bounded absorbing set in $H_0^1\left(\Omega \right)\times L_{\mu }^2(\mathbb{R};H^2\left(\Omega \right)\cup H_0^1\left(\Omega \right))$ is established. For more related works on the long-time behavior of the equation with memory, we refer readers to [8,9,10,11,12,13].

In general, the aforementioned results presuppose that $g\in L^2\left(\Omega \right)$ in order to obtain a bounded absorbing set or an energy estimate within a higher regularity space (such as $H_0^1\left(\Omega \right)\times L_{\mu }^2({\mathbb{R}}^{+},H^2\left(\Omega \right)\cap H_0^1)$). This assumption is pivotal for establishing compactness or asymptotic compactness in a lower regularity space (the original phase space of the semigroup), which is essential for proving the existence of attractors. In this paper, we shall investigate the existence of a global attractor for Equation (1) under the weaker assumption that $g\in H^{-1}\left(\Omega \right)$. We recall that when $g\in H^{-1}\left(\Omega \right)$, there exists a unique weak solution to Equation (1) in $L^2\left(\Omega \right)\times L_{\mu }^2(\mathbb{R};H_0^1\left(\Omega \right))$, where the solution generates a semigroup as the standard reaction–diffusion equation. It is natural to ask whether the global attractor exists in such a space. Note that when $g\in H^{-1}\left(\Omega \right)$, the standard energy estimates derived by using $-\mathsf{\Delta }u$ or $u_t$ as test functions are not in hand. For standard reaction–diffusion equations, to overcome this difficulty, the authors of [14,15] (see also [16,17]) developed the $\omega –$limit compactness method to investigate the existence of a global attractor. Yet, such a method is not applicable for our problem with memory. Actually, it only ensures that the co-dimension of the first component u is sufficiently small after orthogonal decomposition, but does not address the second component ${\eta }^t$. To overcome this difficulty, we adapt the ideas of Chueshov in [18] to use the contractive function method to establish asymptotic smoothness. First, we prove that the semigroup of solutions possesses a bounded absorbing set in the phase space. Then, we show that the elements of this absorbing set satisfy the compression function condition, and thereby obtain the asymptotic compactness of the semigroup, and the existence of the attractor.

2. Some Preliminaries

Assume that the nonlinear term f in Equation (1) satisfies the following conditions:

$$\begin{array}{c}\hfill {\alpha }_1{\left|s\right|}^p-k_1⩽f\left(s\right)s⩽{\alpha }_2{\left|s\right|}^p+k_2,\begin{array}{c}\hfill p⩾2,\end{array}\end{array}$$

(2)

where ${\alpha }_1,{\alpha }_2,k_1,k_2$, l and C are positive constants satisfying the dissipative conditions

$$\begin{array}{c}\hfill f^{\prime }\left(s\right)⩾-l,\forall s\in \mathbb{R},\end{array}$$

(3)

$$\begin{array}{c}\hfill \langle f\left(s_1\right)-f\left(s_2\right),s_1-s_2\rangle ⩽C.\end{array}$$

Let $F\left(s\right)={\int }_0^{s}f\left(\tau \right)d\tau$. It is not difficult to see that there exist positive constants ${\alpha }_3,{\alpha }_4,k_3,\mathrm{and}{\delta }_1$, such that

$$\begin{array}{c}\hfill {\alpha }_3{\left|s\right|}^p-{\delta }_1{\left|s\right|}^2⩽F\left(s\right)s⩽{\alpha }_4{\left|s\right|}^p+k_3{\left|s\right|}^2.\end{array}$$

(4)

Concerning the memory kernel function $k\left(s\right)$, let $\mu \left(s\right)=-k^{\prime }\left(s\right)$. Hypothesize that

$$\begin{array}{c}\hfill \mu \in C^1\begin{array}{c}\hfill \left({\mathbb{R}}^{+}\right)\cap L^1\left({\mathbb{R}}^{+}\right)\end{array},\mu \left(s\right)⩾0,{\mu }^{\prime }\left(s\right)⩽0,\forall s\in {\mathbb{R}}^{+},\end{array}$$

(5)

and there are constants $\delta ,\gamma >0$, such that

$$\begin{array}{c}\hfill {\mu }^{\prime }\left(s\right)+\delta \mu \left(s\right)⩽0,\end{array}$$

(6)

$$\begin{array}{c}\hfill \mu (\infty )=\underset{s\to \infty }{lim}\mu \left(s\right)=0,\end{array}$$

(7)

$$\begin{array}{c}\hfill m_0={\int }_0^{\infty }\mu \left(s\right)ds.\end{array}$$

For the past history of u, denote

$$\begin{array}{c}\hfill {\eta }^t={\eta }^t(x,s):={\int }_0^{s}u(x,t-\tau )d\tau ,\forall s\in {\mathbb{R}}^{+},\end{array}$$

(8)

and

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\eta }_t^t={\displaystyle \frac{\partial }{\partial t}}{\eta }^t,\\ \hfill {\eta }_s^t={\displaystyle \frac{\partial }{\partial s}}{\eta }^t.\end{array}\end{array}$$

The historical variable $u_0(·,-s)$ of u satisfies the following conditions: there exist two positive N and $\sigma \le \delta$ such that

$$\begin{array}{c}\hfill {\int }_0^{\infty }{e}^{-\delta s}\left|\right|u_0(-s){\left|\right|}_0^{2}ds⩽N.\end{array}$$

(9)

From Equations (7) and (8), and utilizing properties of the Lebesgue integral and partial integrals, we get

$$\begin{array}{c}\hfill {\int }_0^{\infty }k\left(s\right)\mathsf{\Delta }u(t-s)ds={\int }_0^{\infty }\mu \left(s\right)\mathsf{\Delta }{\eta }^t\left(s\right)ds.\end{array}$$

Therefore, transforming the memory term in Equation (1) yields the following system

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{u}_t-\mathsf{\Delta }u-{\int }_0^{\infty }\mu \left(s\right)\mathsf{\Delta }{\eta }^t\left(s\right)ds+f\left(u\right)=g,\hfill \\ {\eta }_t^t=-{\eta }_s^t+u,\hfill \end{array}\right.\end{array}\end{array}$$

(10)

with the initial boundary conditions

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{u(x,t)|}_{\partial \Omega }=0,{\eta }^t{(x,s)|}_{\partial \Omega \times {\mathbb{R}}^{+}}=0,t\ge 0,\hfill \\ u(x,0)=u_0\left(x\right),{\eta }^0(x,s)={\int }_0^s{u}_0(x,-\tau )d\tau ,(x,s)\in \Omega \times {\mathbb{R}}^{+}.\hfill \end{array}\right.\end{array}\end{array}$$

(11)

In this manuscript, let $\Omega$ be a fixed domain in ${\mathbb{R}}^n(n⩾3)$. $L^2\left(\Omega \right)$ is denoted by a Hilbert space H, $H_0^1\left(\Omega \right)$ is denoted by V. The norms in H and V will be denoted by $|·|$ and $\parallel ·\parallel$, respectively. Let X be a Hilbert space, and $L_{\mu }^2({\mathbb{R}}^{+};X)$ be a Hilbert space of functions $\phi :\mathbb{R}\to X$ endowed with the inner product

$$\begin{array}{c}\hfill {\langle {\phi }_1,{\phi }_2\rangle }_{\mu ,X}={\int }_0^{\infty }\mu \left(s\right){〈{\phi }_1\left(s\right),{\phi }_2\left(s\right)〉}_{X}ds.\end{array}$$

Let ${\parallel ·\parallel }_{\mu ,X}$ denote the corresponding norm

$$\begin{array}{c}\hfill {\left|\right|\phi \left|\right|}_{\mu ,X}^2={\int }_0^{\infty }{\mu \left(s\right)\left|\right|\phi \left(s\right)\left|\right|}_X^{2}ds.\end{array}$$

In particular, in the case that $X=V$, $X=V\cap H^2\left(\Omega \right)$, we denote the norm as ${\parallel \phi \parallel }_{\mu ,1}$ and ${\parallel \phi \parallel }_{\mu ,2}$, respectively.

Moreover, let

$$\begin{array}{c}\hfill {\mathcal{L}}_1=H\times L_{\mu }^2({\mathbb{R}}^{+},V)\end{array}$$

and

$$\begin{array}{c}\hfill {\mathcal{L}}_2=V\times L_{\mu }^2({\mathbb{R}}^{+};V\cap H^2\left(\Omega \right)).\end{array}$$

with norm

$$\begin{array}{c}\hfill {\parallel z_1\parallel }_{{\mathcal{L}}_1}^2={\parallel (u_1,{\phi }_1)\parallel }_{{\mathcal{L}}_1}^2=\left|u_1\right|+{\parallel {\phi }_1\parallel }_{\mu ,1}^2,\end{array}$$

$$\begin{array}{c}\hfill {\parallel z_2\parallel }_{{\mathcal{L}}_2}^2={\parallel (u_2,{\phi }_2)\parallel }_{{\mathcal{L}}_2}^2=\left|u_2\right|+{\parallel {\phi }_2\parallel }_{\mu ,2}^2,\end{array}$$

for any $z_1=(u_1,{\phi }_1)\in {\mathcal{L}}_1$, and $z_2=(u_2,{\phi }_2)\in {\mathcal{L}}_2$.

Lemma 1

([19]). Let X be a Hilbert space, $I=[0,T]$, $\forall T\ge 0$. Assume the memory kernel $\mu \left(s\right)$ satisfies (4) and (5). Then, for any ${\eta }^t\in C(I,L_{\mu }^2({\mathbb{R}}^{+},X))$, we have

$$\begin{array}{c}\hfill {\langle {\eta }^t,{\eta }_s^t\rangle }_{\mu ,X}⩾{\displaystyle \frac{\delta }{2}}\left|\right|{\eta }^t{\left|\right|}_{\mu ,X}^2,\end{array}$$

where δ is from (6).

Proof. 

Integrating by parts in s and using (6), it yields

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\langle {\eta }^t,{\eta }_s^t\rangle }_{\mu ,X}& ={\int }_0^{\infty }\mu d\left({\displaystyle \frac{1}{2}}{\int }_{\Omega }\left|\right|\nabla {\eta }^t\left(s\right){\left|\right|}^{2}dx\right)\hfill \\ \hfill & ={\left[{\displaystyle \frac{1}{2}}\mu \left(s\right)\left|\right|\nabla {\eta }^t\left(s\right){\left|\right|}^2\right]}_0^{\infty }-{\displaystyle \frac{1}{2}}{\int }_0^{\infty }{\mu }^{\prime }\left(s\right)\left|\right|\nabla {\eta }^t\left(s\right){\left|\right|}^{2}ds\hfill \\ \hfill & =-{\displaystyle \frac{1}{2}}{\int }_0^{\infty }{\mu }^{\prime }\left(s\right)\left|\right|\nabla {\eta }^t\left(s\right){\left|\right|}^{2}ds\hfill \\ \hfill & ⩾{\displaystyle \frac{\delta }{2}}\left|\right|{\eta }^t\left(s\right){\left|\right|}_{\mu ,X}^2.\hfill \end{array}\end{array}$$

□

We now recall the definitions of the asymptotic smoothness of the semigroup and the existence theorem of attractors, which will be used later.

Definition 1

([18], Definition 2.2.1). Let ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ be a semigroup on a complete metric space X. ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ is said to be asymptotically compact if the following Ladyzhenskaya condition holds: for any bounded set B in X such that the tail ${\gamma }^{\tau }\left(B\right):={\cup }_{t⩾\tau }S\left(t\right)B$ is bounded for some $\tau ⩾0$, we have that any sequence of the form $\left\{S\left(t_n\right)x_n\right\}$ with $x_n\in B$ and $t_n\to \infty$ is relatively compact.

Definition 2

([18], Definition 2.2.1). Let ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ be a semigroup on a complete metric space X, and let $(X,S_t)$ be the corresponding dynamical system. A closed set $B\subset X$ is said to be absorbing for ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ if for any bounded set $D\subset X$ there exists $t_0\left(D\right)$ such that $S\left(t\right)D\subset B$ for all $t⩾t_0\left(D\right)$.

Definition 3

([18], Definition 2.2.1). Let ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ be a semigroup on a complete metric space X. ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ is said to be asymptotically smooth if the following Hale condition is valid: for every bounded set D such that $S\left(t\right)D\subset D$ for $t>0$ there exists a compact set K in the closure $\overline{D}$ of D, such that $S\left(t\right)D$ converges uniformly to K in the sense that

$$\begin{array}{c}\hfill \underset{t\to +\infty }{lim}{d}_X\left\{S\left(t\right)D\right|K\}=0,\mathit{where}{d}_X\left\{A\right|B\}=\underset{x\in A}{sup}dis{t}_X(x,B).\end{array}$$

Lemma 2

([18], Proposition 2.2.4). A semigoup ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ in some metric space X is asymptotically compact if and only if it is asymptotically smooth.

Lemma 3

([18], Theorem 2.3.5). Let $(X,S(t\left)\right)$ be a dissipative asymptotically compact dynamical system on a complete metric space X. Then, $S\left(t\right)$ possesses a unique compact global attractor E such that

$$\begin{array}{c}\hfill E=\omega \left(B_0\right)=\bigcap _{t>0}\overline{\bigcup _{\tau \ge t_1}S\left(\tau \right)B_0},\end{array}$$

for every bounded absorbing set $B_0$ and

$$\begin{array}{c}\hfill \underset{t\to +\infty }{lim}(d_X\left\{S\left(t\right)B_0\right|E\}+d_X\left\{E\right|S\left(t\right)B_0\})=0.\end{array}$$

where, as above, $d_X\left\{A\right|B\}={sup}_{x\in A}dis{t}_X(x,B)$. Moreover, if there exists a connected absorbing bounded set, then E is connected.

Thanks to Lemmas 2 and 3, the existence of the attractor can be achieved through asymptotic smoothness. The next lemma provides a sufficient condition for verifying the asymptotic smoothness using the contractive functions method.

Lemma 4

([18], Theorem 2.2.17). Let ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ be a semigroup on a complete metric space X. Assume that for any bounded forward invariant set B in X and for any $\epsilon >0$ there exists $T\equiv T(\epsilon ,B)$ such that

$$\begin{array}{c}\hfill dist(S\left(T\right)y_1,S\left(T\right)y_2)⩽\epsilon +{\Psi }_{\epsilon ,B,T}(y_1,y_2),y_i\in B,\end{array}$$

where ${\Psi }_{\epsilon ,B,T}(y_1,y_2)$ is a functional defined on $B\times B$ such that

$$\begin{array}{c}\hfill \underset{m⟶\infty }{lim\; inf}\underset{n⟶\infty }{lim\; inf}{\Psi }_{\epsilon ,B,T}(y_n,y_m)=0,\mathit{for}\mathit{every}\mathit{sequence}{y}_n\subset B.\end{array}$$

Then, ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ is asymptotically smooth.

A function ${\Psi }_{\epsilon ,B,T}(y_1,y_2)$ satisfying (2.23) is said to be a contractive function on B, and $Cont\left(B\right)$ denotes the set of all contractive functions on B.

Additionally, the following lemma is useful in subsequent disccusions.

Lemma 5

([20], Aubin–Dubinskii–Lions Lemma). Assume that $X\subset Y\subset Z$ is a triple of Banach spaces such that $X\hookrightarrow \hookrightarrow Y\hookrightarrow Z$.

(1) 

Let $W_1=\{u\in L^p(a,b;X)|u_t\in L^q(a,b;Z)\}$ for some $1\le p\le \infty$ and $q\ge 1$. Here, $u_t$ denotes the derivative of u in the distributional sense. Then, $W_1\hookrightarrow \hookrightarrow L^p(a,b;Y).$ If $q>1$, then $W_1\hookrightarrow \hookrightarrow C(a,b;Z);$

(2) 

Let $W_2=\{u\in L^{\infty }(a,b;X)|u_t\in L^r(a,b;Z)\}$ for some $r>1$, then $W_2\hookrightarrow \hookrightarrow C(a,b;Y)$.

3. Asymptotical Smoothness and Global Attractors

For convenience, denote g as $h+D_i{g}^i$($=h+{\Sigma }_{i=1}^n{D}_i{g}^i$), where $h,g^i\in L^2\left(\Omega \right)$. As presented in the previous section, the Dafermos transformation allows us to reformulate the original problem (1) as a dynamical system

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{u}_t-\mathsf{\Delta }u-{\int }_0^{\infty }\mu \left(s\right)\mathsf{\Delta }{\eta }^t\left(s\right)ds+f\left(u\right)=h+D_i{g}^i,\hfill \\ {\eta }_t^t=-{\eta }_s^t+u,\hfill \end{array}\right.\end{array}\end{array}$$

(12)

with initial boundary conditions

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{u(x,t)|}_{\partial \Omega }=0,{\eta }^t{(x,s)|}_{\partial \Omega \times {\mathbb{R}}^{+}}=0,t\ge 0,\hfill \\ u(x,0)=u_0\left(x\right),{\eta }^0(x,s)={\int }_0^s{u}_0(x,-\tau )d\tau ,(x,s)\in \Omega \times {\mathbb{R}}^{+}.\hfill \end{array}\right.\end{array}\end{array}$$

(13)

This section addresses the existence of a global attractor for the Equation (12) under the assumptions (2)–(9). By [21] (see also [7]), we have the existence and uniqueness of weak solutions.

Lemma 6.

Assume that (2)–(9) hold and $h,g^i\in L^2\left(\Omega \right)$. Then, for any $T>0$ and $z_0=(u_0,{\eta }^0)\in {\mathcal{L}}_1$, there exists a unique weak solution $z=(u,{\eta }^t)$ for the Equation (12) with the initial conditions (13), satisfying

$$\begin{array}{cc}\hfill u\in L^{\infty }(0,T;H)& \cap L^2(0,T;V)\cap L^P(0,T;L^P\left(\Omega \right)),\hfill \\ \hfill {\eta }^t& \in L^{\infty }(0,T;L_{\mu }^2({\mathbb{R}}^{+};V),\hfill \\ \hfill u_t& \in L^2(0,T;H^{-1}\left(\Omega \right)).\hfill \end{array}$$

Furthermore,

$$\begin{array}{c}\hfill z\in C(0,T;H),\forall T>0.\end{array}$$

The above lemma insures the existence of a solution semigroup ${\left\{S\left(t\right)\right\}}_{t⩾0}$ in ${\mathcal{L}}_1$

$$\begin{array}{cc}\hfill S\left(t\right)& :{\mathbb{R}}^{+}\times {\mathcal{L}}_1\to {\mathcal{L}}_1,\hfill \\ \hfill & S\left(t\right)z_0=z\left(t\right).\hfill \end{array}$$

Remark 1.

Actually, it suffices for the above Lemma 6 to hold when $h,g^i\in L^2\left(\Omega \right)$ (i.e., $g=h+D_i{g}^i\in H^{-1}\left(\Omega \right))$. It is only necessary for $g\in L^2\left(\Omega \right)$ when proving that the solutions have higher regularity, such as if $z_0\in {\mathcal{L}}_2$, then

$$\begin{array}{c}u\in L^{\infty }(0,T;V)\cap L^2(0,T;V\cap H^2\left(\Omega \right)),\\ {\eta }^t\in L^{\infty }(0,T;L_{\mu }^2({\mathbb{R}}^{+};V\cap H^2\left(\Omega \right)),\mathit{and}z\in C(0,T;V).\end{array}$$

The following lemma provides the existence of a bounded absorbing set in ${\mathcal{L}}_1=H\times L_{\mu }^2({\mathbb{R}}^{+},V)$.

Lemma 7.

Assume that the conditions in Lemma 6 hold. Then, there exist positive constants C and $t_0=t_0\left(\right||z_0{\left|\right|}_{{\mathcal{L}}_1})$ such that,

$$\begin{array}{c}\hfill \left|\right|S\left(t\right)z_0{\left|\right|}_{{\mathcal{L}}_1}\le C,\forall t\ge t_0,\end{array}$$

Therefore,

$$\begin{array}{c}\hfill E_0=\{z\in {\mathcal{L}}_1{:\left|\right|z\left|\right|}_{{\mathcal{L}}_1}\le C\}\end{array}$$

is a bounded absorbing set for the semigroup ${S\left(t\right)}_{t\ge t_0}$ in ${\mathcal{L}}_1$.

Proof. 

Multiplying Equation (12) by u and integrating over $\Omega$, we get

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\left|u\right|}^2+{\left|\right|u\left|\right|}^2+{\int }_0^{\infty }\mu \left(s\right)\langle -\mathsf{\Delta }{\eta }^t\left(s\right),u\left(t\right)\rangle ds+\langle f\left(u\right),u\rangle =\langle h,u\rangle +\langle D_i{g}^i,u\rangle .\end{array}$$

(14)

For the third term on the left-hand side, we use (8), where $u={\eta }_t^t+{\eta }_s^t$ to deduce that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\int }_0^{\infty }\mu \left(s\right)\langle -\mathsf{\Delta }{\eta }^t\left(s\right),u\left(t\right)\rangle ds& ={\int }_0^{\infty }\mu \left(s\right)\langle -\mathsf{\Delta }{\eta }^t\left(s\right),{\eta }_t^t\rangle ds+{\int }_0^{\infty }\mu \left(s\right)\langle -\mathsf{\Delta }{\eta }^t\left(s\right),{\eta }_s^t\rangle ds\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\int }_0^{\infty }\mu \left(s\right)\left|\right|{\eta }^t{\left(s\right)\left|\right|}^{2}ds+{\int }_0^{\infty }\mu \left(s\right)\langle \nabla {\eta }^t\left(s\right),\nabla {\eta }_s^t\left(s\right)\begin{array}{c}\hfill \rangle \end{array}ds\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\left|\right|{\eta }^t{\left|\right|}_{\mu ,1}^2+{\langle {\eta }^t,{\eta }_s^t\rangle }_{\mu ,1}.\hfill \end{array}\end{array}$$

(15)

From Lemma 1, ${\langle {\eta }^t,{\eta }_s^t\rangle }_{\mu ,1}⩾\frac{\delta }{2}\left|\right|{\eta }^t{\left|\right|}_{\mu ,1}^2$. We therefore obtain that

$$\begin{array}{c}\hfill {\int }_0^{\infty }\mu \left(s\right)\langle -\mathsf{\Delta }{\eta }^t\left(s\right),u\left(t\right)\rangle ds⩾{\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\left|\right|{\eta }^t{\left|\right|}_{\mu ,1}+{\displaystyle \frac{\delta }{2}}\left|\right|{\eta }^t{\left|\right|}_{\mu ,1}.\end{array}$$

(16)

On the other hand, note that

$$\begin{array}{c}\hfill {\langle f\left(u\right),u)\rangle ⩾\alpha \left|u\right|}_p^p-k_1{\left|\Omega \right|}^2,\end{array}$$

(17)

$$\begin{array}{c}\hfill \langle h,u\rangle ⩽{\displaystyle \frac{1}{\lambda }}{\left|h\right|}^2+{\displaystyle \frac{\lambda }{4}}{\left|u\right|}^2,\end{array}$$

(18)

where $\lambda$ is the Sobolev constant, which satisfies the condition ${\left|\right|u\left|\right|}^2⩾\lambda {\left|u\right|}^2$. Moreover,

$$\begin{array}{c}\hfill \langle D_i{g}^i,u\rangle =-\int \tilde{g}·\nabla u⩽|\tilde{g}{|}^2+{\displaystyle \frac{1}{4}}{\left|\right|u\left|\right|}^2.\end{array}$$

(19)

where $\tilde{g}=(g^1,g^2,···,g^n)$, $|\tilde{g}{|}^2={\sum }_{i=1}^n{|g^i|}^2$. By taking (15)–(19) into (14), we get

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\left|u\right|}^2+{\left|\right|u\left|\right|}^2+{\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\left|\right|{\eta }^t{\left|\right|}_{\mu ,1}^2+{\displaystyle \frac{\delta }{2}}\left|\right|{\eta }^t{\left|\right|}_{\mu ,1}^2+\alpha {\left|u\right|}_p^p\hfill \\ \hfill ⩽& k_1{\left|\Omega \right|}^2+{\displaystyle \frac{1}{\lambda }}{\left|h\right|}^2+{\displaystyle \frac{\lambda }{4}}{\left|\right|u\left|\right|}_2^2+{\left|\tilde{g}\right|}^2+{\displaystyle \frac{1}{4}}{\left|\right|u\left|\right|}^2.\hfill \end{array}$$

By substituting ${\left|\right|u\left|\right|}^2⩾\lambda {\left|u\right|}_2^2$ into the formula and rearranging, it follows that

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\left(\right|u|}^2+\left|\right|{\eta }^t{\left|\right|}_{\mu ,1}^2{)+(\left|\right|u\left|\right|}^2-{\displaystyle \frac{\lambda }{2}}{\left|u\right|}^2+{\displaystyle \frac{\delta }{2}}\left|\right|{\eta }^t{\left|\right|}_{\mu ,1}^2)⩽{\displaystyle \frac{\tilde{C}\left(\right|h|,|\tilde{g}|,\lambda ,|\Omega \left|\right)}{2}}.\end{array}$$

Taking $\gamma =min\{\lambda ,\delta \}$, we have

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}{\left(\right|u|}^2+\left|\right|{\eta }^t{\left|\right|}_{\mu ,1}^2{)+\gamma (\left|u\right|}^2+\left|\right|{\eta }^t{\left|\right|}_{\mu ,1}^2)⩽\tilde{C}\left(\right|h|,|\tilde{g}|,\lambda ,|\Omega \left|\right),\end{array}$$

for which Gronwall’s lemma implies that

$$\begin{array}{c}\hfill {\left|u\left(t\right)\right|}^2+\left|\right|{\eta }^t{\left(s\right)\left|\right|}_{\mu ,1}^2⩽\left(\right|u_0{\left(x\right)|}^2+\left|\right|{\eta }^0{\left(s\right)\left|\right|}_{\mu ,1}^2)e^{-\gamma t}+{\displaystyle \frac{1}{\gamma }}\tilde{C}.\end{array}$$

Therefore,

$$\begin{array}{c}\hfill {\left|\right|z\left(t\right)\left|\right|}_{{\mathcal{L}}_1}⩽\left|\right|z_0{\left|\right|}_{{\mathcal{L}}_1}{e}^{-\gamma t}+{\displaystyle \frac{\tilde{C}}{\gamma }}.\end{array}$$

Taking $t_0=\frac{1}{\gamma }ln\frac{\gamma }{\tilde{C}}$, $C^2=\frac{2}{\gamma }\tilde{C}$, we have

$$\begin{array}{c}\hfill {\left|\right|z\left(t\right)\left|\right|}_{{\mathcal{L}}_1}⩽C,\forall t⩾t_0.\end{array}$$

□

Next, we will prove that the semigroup generated by the weak solution of Equation (12) satisfies the conditions of Lemma 4, thereby obtaining the asymptotic smoothness of the semigroup.

Lemma 8.

Assume that the conditions in Lemma 6 hold. Then, ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ is asymptotically smooth.

Proof. 

It suffices to prove that the statement in Lemma 4 holds true for the absorbing set $E_0$ constructed in Lemma 7. To this end, take any $(u_0^1,{\eta }_1^0),(u_0^1,{\eta }_1^0)\in {\mathcal{L}}_1$ and set $(u_p\left(t\right),{\eta }_p^t)=S_t(u_0^p,{\eta }_p^0),p=1,2$. Then, we have

$$\begin{array}{c}\hfill {\left(u_p\right)}_t-\mathsf{\Delta }{u}_p-{\int }_0^{\infty }\mu \left(s\right)\mathsf{\Delta }{\eta }_p^t\left(s\right)ds+f\left(u_p\right)=h+D_i{g}^i,(p=1,2).\end{array}$$

Let $w=u_1-u_2$, $\eta ={\eta }_1^t-{\eta }_2^t$; it is obvious that

$$\begin{array}{c}\hfill w_t-\mathsf{\Delta }w-{\int }_0^{\infty }\mu \left(\mathsf{\Delta }s\right)\mathsf{\Delta }{\eta }^t\left(s\right)ds+f\left(u_1\right)-f\left(u_2\right)=0.\end{array}$$

(20)

Multiplying the Equation (20) by w, we deduce that

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\left|w\right|}^2+{|\nabla w|}^2+{\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\left|\right|{\eta }^t{\left|\right|}_{\mu ,1}^2+{\displaystyle \frac{\delta }{2}}\left|\right|{\eta }^t{\left|\right|}_{\mu ,1}^2⩽\langle f\left(u_2\right)-f\left(u_1\right),u_1-u_2\rangle .\end{array}$$

From (3), it follows that

$$\begin{array}{c}\hfill \langle f\left(u_2\right)-f\left(u_1\right),u_1-u_2\rangle \le l{|u_1-u_2|}^2=l{\left|w\right|}^2.\end{array}$$

Note that

$${\left|w\right|}^2⩽{\displaystyle \frac{1}{{\lambda }_1}}{|\nabla w|}^2.$$

Therefore

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}{(|w|}^2+{\parallel {\eta }^t\parallel }_{\mu ,1}^2)+(2{\lambda }_1+\delta )({\left|w\right|}^2+{\parallel {\eta }^t\parallel }_{\mu ,1}^2){⩽l\left|w\right|}^2.\end{array}$$

By Gronwall’s lemma, we get for all $t\ge 0$ that

$$\begin{array}{cc}\hfill {|w(t)|}^2+{\parallel {\eta }^t\parallel }_{\mu ,1}^2& ⩽e^{-(2{\lambda }_1+\delta )t}{(|w(0)|}^2+\parallel {\eta }^0{\parallel }_{\mu ,1}^2)+l{\int }_0^t{e}^{-(2{\lambda }_1+\delta )(t-s)}{|w(s)|}^{2}ds\hfill \\ \hfill & ⩽e^{-(2{\lambda }_1+\delta )t}(|u_0^1-u_0^2{|}^2+{\parallel {\eta }_1^0-{\eta }_2^0\parallel }_{\mu ,1}^2)+l{\int }_0^t{|w(s)|}^{2}ds,\hfill \end{array}$$

(21)

where $(u_0^1,{\eta }_1^0),(u_0^2,{\eta }_2^0)\in B_0$ are the initial data of solutions $u_1,u_2$, and $B_0$ is the absorbing set. For any $\epsilon >0$, we choose $T=T(\epsilon ,B_0)>0$ such that

$$e^{-\frac{(2{\lambda }_1+\delta )}{2}T}{(|u_0^1-u_0^2{|}^2+{\parallel {\eta }_1^0-{\eta }_2^0\parallel }_{\mu ,1}^2)}^{\frac{1}{2}}⩽C_{B_0}{e}^{-\frac{(2{\lambda }_1+\delta )}{2}T}<\epsilon .$$

From (21), it follows that

$$\parallel S(T)(u_0^1,{\eta }_1^0)-S(T)(u_0^2,{\eta }_2^0)\parallel ⩽\epsilon +{\Psi }_T((u_0^1,{\eta }_1^0),(u_0^2,{\eta }_2^0)),$$

where

$${\Psi }_T\left((u_0^1,{\eta }_1^0),(u_0^2,{\eta }_2^0)\right)=\sqrt{l}{\left({\int }_0^T{|u_1\left(s\right)-u_2\left(s\right)|}^{2}ds\right)}^{\frac{1}{2}}.$$

Next, we show that ${\Psi }_T(·,·)\in \mathrm{Cont}\left(B_0\right)$.

By (2.1), we know that

$$\left|f\left(s\right)\right|⩽c_0{(1+|s|}^{p-1}),\forall s\in \mathbb{R}.$$

As a result, $u\in L^p([0,T]\times \Omega )$ implies that $f\left(u\right)\in L^q([0,T]\times \Omega )$ with ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1$. Specifically,

$$\begin{array}{cc}\hfill {\parallel f\left(u\right)\parallel }_{L^q([0,T]\times \Omega )}^q& ⩽c_0{\int }_0^T{\int }_{\Omega }{(1+{\left|u\right|}^{p-1})}^{q}dxdt\hfill \\ \hfill & ⩽C{\int }_0^T{\int }_{\Omega }{(1+|u|}^{q(p-1)})dxdt\hfill \\ \hfill & ⩽CT\left|\Omega \right|+C{\parallel u\parallel }_{L^p([0,T]\times \Omega )}^p.\hfill \end{array}$$

From Lemma 6, it follows that $u_t\in L^2(0,T;H^{-1})\cap L^q(0,T;L^q)$. Note that both $L^2(0,T;H^{-1})$ and $L^q(0,T;L^q)$ are continuous and included in $L^q(0,T;H^{-s})$ (with $q<2$ since $p>2$ and s are chosen such that $H^s\hookrightarrow L^p$). Thus, for any sequence ${\left\{(u_0^n,{\eta }_n^0)\right\}}_{n⩾1}\subset B_0$, setting $(u^n\left(t\right),{\eta }_n^t)=S\left(t\right)(u_0^n,{\eta }_n^0)$, it follows that

$$\begin{array}{c}\hfill u^n\in L^2(0,T;H_0^1),u_t^n\in L^q(0,T;H^{-s}).\end{array}$$

Since

$$H_0^1\hookrightarrow \hookrightarrow L^2\hookrightarrow H^{-s}.$$

By the Aubin–Lions Lemma 5, there exists a subsequence ${\left\{(u_0^{n_k},{\eta }_{n_k}^0)\right\}}_{k⩾1}\subset {\left\{(u_0^n,{\eta }_n^0)\right\}}_{n⩾1}$ such that

$$\begin{array}{c}\hfill \underset{i\to \infty }{lim}\underset{j\to \infty }{lim}{\Psi }_T\left((u_0^{n_i},{\eta }_{n_i}^0),(u_0^{n_j},{\eta }_{n_j}^0)\right)=\underset{i\to \infty }{lim}\underset{j\to \infty }{lim}\sqrt{l}{\left({\int }_0^T{|u_{n_i}\left(s\right)-u_{n_j}\left(s\right)|}^{2}ds\right)}^{\frac{1}{2}}=0.\end{array}$$

This implies that ${\Psi }_T(·,·)\in Cont\left(B_0\right)$ and the proof is complete by Lemma 4. □

Based on Lemmas 2, 3 and 8, we are ready to derive the main result regarding the existence of the global attractor immediately.

Theorem 1.

Assume that (2)–(9) hold and $h,g^i\in L^2\left(\Omega \right)$. The semigroup ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ generated by Equation (12) possesses a global attractor $\mathcal{A}$ in ${\mathcal{L}}_1=H\times L_{\mu }^2({\mathbb{R}}^{+},V)$.