Uniform attractors for the non-autonomous reaction-diffusion equations with delays

Abstract

In this paper, we consider a non-autonomous reaction-diffusion equation with hereditary effects and the nonlinearity f satisfying the polynomial growth of arbitrary $p - 1$ ( $p ⩾ 2$ ) order. We employ the asymptotic a priori estimate method (see (J. Differential Equations 223 (2006) 367–399)) to our problem and establish an existence criterion for the $(C_{L^{2} (Ω)}, C_{L^{p} (Ω)})$ -uniform (w.r.t $σ \in Σ$ ) attractors (see Theorem 2.18). Then, we obtain the $(C_{L^{2} (Ω)}, C_{L^{p} (Ω)})$ and $(C_{L^{2} (Ω)}, C_{H_{0}^{1} (Ω)})$ -uniform (w.r.t $σ \in Σ$ ) attractors by applying the existence criterion and the uniform (w.r.t $σ \in Σ$ ) Condition (C) respectively.

Keywords

Reaction-diffusion equations delays uniform attractors

1. Introduction

Delays in differential systems are used as mathematical models to describe of the dynamics behavior depending on past events. For this reason, delay differential equations (DDE for short) are received extensive attention and widely applied in Physical and Chemical processes, Engineering systems, Biological systems and Communication systems, etc., (see [17] for details). In the field of Mathematics, people mainly pay attention to the well-posedness and the long-time behavior of solutions for the DDE. For the well-posedness of solutions and the long-time behavior about DDE, there exists a great deal of literature, see for instance [16] and [3–8,11–15,18,19,22,23,32,33].

Now, we state our problem properly.

Let $Ω \subset R^{N}$ ( $N ⩾ 3$ ) be a bounded domain with smooth boundary, we consider the long-time behavior of solutions for the following reaction-diffusion equations with delays: $\begin{matrix} (1.1) & \{\begin{matrix} \partial_{t} u - Δ u = f (u) + g (u_{t}) + k (t) & in Ω \times (τ, \infty), \\ u (x, t) = 0 & on \partial Ω \times (τ, \infty), \\ u (x, τ + θ) = ϕ (x, θ), & x \in Ω, θ \in [- h, 0], \end{matrix} \end{matrix}$ where $τ \in R$ , g is a operator acting on the solutions containing some hereditary characteristic (assumptions on g are given below), the time-dependent external force term $k (\cdot) \in L_{loc}^{2} (R; L^{2} (Ω))$ , $ϕ \in C ([- h, 0]; L^{2} (Ω))$ is the initial data, $h (> 0)$ is the length of the delay effects, and for each $t ⩾ τ$ , we denote by $u_{t}$ the function defined in $[- h, 0]$ with $u_{t} (θ) = u (t + θ)$ , $θ \in [- h, 0]$ .

We will denote by $C_{X}$ the Banach space $C ([- h, 0]; X)$ , equipped with the sup-norm. For an element $u \in C_{X}$ , its norm will be written as $‖ u ‖_{C_{X}} = {max}_{t \in [- h, 0]} ‖ u (t) ‖_{X}$ .

For the operator g, similar to that in [11,31], we will assume that $g (\cdot) : C_{L^{2} (Ω)} \to L^{2} (Ω)$ and:

$g (0) = 0$ ;

there exists a constant $L_{g} > 0$ such that for all $ξ_{1}, ξ_{2} \in C_{L^{2} (Ω)}$ , $\begin{array}{l} {‖ g (ξ_{1}) - g (ξ_{2}) ‖}_{2} ⩽ L_{g} ‖ ξ_{1} - ξ_{2} ‖_{C_{L^{2} (Ω)}} . \end{array}$

For the nonlinearity $f \in C (R; R)$ , we make the following classical assumptions (e.g., see [1,26,30]): $\begin{matrix} (1.2) & (f (u) - f (v)) (u - v) ⩽ l {(u - v)}^{2}, \end{matrix}$ and $\begin{matrix} (1.3) & - c_{0} - c_{1} | u |^{p} ⩽ f (u) u ⩽ c_{0} - c_{2} | u |^{p}, p ⩾ 2 \end{matrix}$ for some positive constants $c_{0}$ , $c_{1}$ , $c_{2}$ and all $u, v \in R$ .

Let $F (u) = \int_{0}^{u} f (r) d r$ , then from (1.3), there exist constants ${\tilde{c}}_{i} > 0$ ( $i = 0, 1, 2$ ) such that $\begin{array}{l} (1.4) & - {\tilde{c}}_{0} - {\tilde{c}}_{1} | u |^{p} ⩽ F (u) ⩽ {\tilde{c}}_{0} - {\tilde{c}}_{2} | u |^{p}, \forall u \in R . \end{array}$

The long time behavior of the solutions of the reaction-diffusion equations has been considered by many researchers; see, for example, [26,30] and [19,24,25,27,28,32,34,35].

For the case without delays, especially for the autonomous case, many results associated with this problem are mainly concentrated upon the existence, regularity and dimension of the global attractor in $L^{2} (Ω)$ ; e.g., see [24–26,30]. In 2006, the authors of [35] introduced the asymptotic a priori estimate methods and proved the existence of the global attractor in $L^{p} (Ω)$ and $H_{0}^{1} (Ω)$ with the external force term $g (x) \in H^{- 1} (Ω)$ . Furthermore, they have also obtained the existence of the global attractor in $L^{2 p - 2} (Ω)$ and $H_{0}^{1} (Ω) \cap H^{2} (Ω)$ under the assumption that the external forcing $g (x) \in L^{2} (Ω)$ . In the following, for the non-autonomous case, the authors of [27,28] have proved the existence of the uniform attractors in $L^{p} (Ω)$ and $H_{0}^{1} (Ω)$ respectively with the external forcing $g (x, t) \in L_{n}^{2} (R; L^{2} (Ω))$ (see [21] for details). Moreover, in [34], the author introduced the weakly normal, strongly normal, space regular and time regular external forces. As the applications, they obtained the uniform attractors in corresponding phase spaces for the 2D Navier-stokes equations, weakly damped wave equations and reaction-diffusion equations in unbounded domain by combining with the energy method (see [2] for details).

For the reaction-diffusion equations with delays, in [19], the authors have considered the linear case and obtained the existence and the structure of the uniform attractors in $C_{L^{2} (Ω)}$ (see Theorem 4.3 and Proposition 4.1 in [19]) with the external force term $g (x, t) \in L_{c}^{2} (R; L^{2} (Ω))$ (see [10] for details). In [32], Wang and Kloeden have considered the multi-valued process of the linear case and obtained the the uniform attractors in $C_{H_{0}^{1} (Ω)}$ under the assumption that the external forcing $g (x, t) \in L_{n}^{2} (R; L^{2} (Ω))$ .

In this paper, we consider the existence of the uniform attractors in $C_{L^{2} (Ω)}$ , $C_{L^{p} (Ω)}$ ( $p > 2$ ) and $C_{H_{0}^{1} (Ω)}$ for the non-autonomous reaction-diffusion equations with delays and the nonlinearity f satisfying the polynomial growth of arbitrary $p - 1$ ( $p ⩾ 2$ ) order under the assumption that the external forcing $k (x, t) \in L_{n}^{2} (R; L^{2} (Ω))$ . For our problem, we will confront two main difficulties when we verify the existence of the uniform attractors in $C_{L^{p} (Ω)}$ ( $p ⩾ 2$ ) and $C_{H_{0}^{1} (Ω)}$ . One difficulty is that the nonlinearity f satisfies the polynomial growth of arbitrary $p - 1$ ( $p ⩾ 2$ ) order, which leads to the fact that the Sobolev embedding is not any longer compact. The other difficulty is that our problem contains delay term $g (u_{t})$ , which makes that the phase space is the Banach space $C_{X}$ rather than X. In the Banach space $C_{X}$ , the already existing methods and techniques for verifying the compactness of the family of processes ${U_{σ} (t, τ)}_{σ \in Σ}$ are not any longer valid.

In order to overcome the above difficulties, we firstly verify the uniformly (w.r.t $σ \in Σ$ ) asymptotic compactness in $C_{L^{2} (Ω)}$ for the family of processes ${U_{σ} (t, τ)}_{σ \in Σ}$ by using the technique in [20] and then combine with the idea of bi-spaces to obtain $(C_{L^{2} (Ω)}, C_{L^{2} (Ω)})$ -uniform (w.r.t $σ \in Σ$ ) attractors (see Theorem 3.7). Then, we obtain the $(C_{L^{2} (Ω)}, C_{L^{p} (Ω)})$ -uniform (w.r.t $σ \in Σ$ ) attractors (see Theorem 3.10) by applying the existence criterion (see Theorems 2.18). Finally, we obtain the $(C_{L^{2} (Ω)}, C_{H_{0}^{1} (Ω)})$ -uniform (w.r.t $σ \in Σ$ ) attractors by employing the uniform (w.r.t $σ \in Σ$ ) Condition (C) (see Theorem 3.12).

The outline of the paper is as follows. In Section 2, we give some notions and results about non-autonomous systems and establish the existence criterion for the uniform (w.r.t $σ \in Σ$ ) attractors in $(C_{L^{2} (Ω)}, C_{L^{p} (Ω)})$ ( $p > 2$ ); In Section 3, we prove the existence of the $(C_{L^{2} (Ω)}, C_{L^{2} (Ω)})$ , $(C_{L^{2} (Ω)}, C_{L^{p} (Ω)})$ and $(C_{L^{2} (Ω)}, C_{H_{0}^{1} (Ω)})$ -uniform (w.r.t $σ \in Σ$ ) attractors for the family of processes ${U_{σ} (t, τ)}_{σ \in Σ}$ .

2. Preliminaries and abstract results

2.1. Preliminaries

In this subsection, we recall some basic concepts about non-autonomous systems, which is similar to those in [9,10,21] and the references therein.

The space of translation bounded functions (see [10]) in $L_{loc}^{2} (R; L^{2} (Ω))$ is $\begin{array}{l} L_{b}^{2} (R; L^{2} (Ω)) = {g \in L_{loc}^{2} (R; L^{2} (Ω)) : sup_{t \in R} \int_{t}^{t + 1} \int_{Ω} {| g (x, s) |}^{2} d x d s < \infty} . \end{array}$

The space of translation compact functions (see [10]) in $L_{loc}^{2} (R; L^{2} (Ω))$ is $\begin{array}{l} L_{c}^{2} (R; L^{2} (Ω)) = & {g \in L_{loc}^{2} (R; L^{2} (Ω)) : For any interval [t_{1}, t_{2}] \subset R, \\ {g (x, h + s) : h \in R} |_{[t_{1}, t_{2}]} is precompact in L^{2} (t_{1}, t_{2}; L^{2} (Ω))} . \end{array}$

Let X be a complete metric space and Σ be a parameter set.

The operators ${U_{σ} (t, τ)}_{σ \in Σ}$ are said to be a family of processes in X with symbol space Σ if, for any $σ \in Σ$ , $\begin{array}{l} U_{σ} (t, s) U_{σ} (s, τ) = U_{σ} (t, τ) \forall t ⩾ s ⩾ τ, τ \in R, \\ U_{σ} (τ, τ) = Id (identity) \forall τ \in R . \end{array}$

Let ${T (s)}_{s ⩾ 0}$ be the translation semigroup on Σ. We say that a family of processes ${U_{σ} (t, τ)}_{σ \in Σ}$ , satisfies the translation identity if $\begin{array}{l} (2.1) & U_{σ} (t + s, τ + s) = U_{T (s) σ} (t, τ) \forall σ \in Σ, t ⩾ τ, τ \in R, s ⩾ 0 . \\ (2.2) & T (s) Σ = Σ \forall s ⩾ 0 . \end{array}$

Let Σ be a parameter set, X, Y are two Banach spaces, $Y ↪ X$ continuously. ${U_{σ} (t, τ)}$ , $t ⩾ τ$ , $τ \in R$ , $σ \in Σ$ is a family of processes in Banach space X. Denote by $B (X)$ the set of all bounded subsets of X.

Definition 2.1.
A set $B_{0} \in B (Y)$ is said to be an $(X, Y)$ -uniformly (w.r.t $σ \in Σ$ ) absorbing set for the family of processes ${U_{σ} (t, τ)}_{σ \in Σ}$ if, for any $τ \in R$ , and $B \in B (X)$ there exists $t_{0} = t_{0} (τ, B) ⩾ τ$ such that $\begin{array}{l} ⋃_{σ \in Σ} U_{σ} (t, τ) B \subset B_{0} for all t ⩾ t_{0} . \end{array}$
Definition 2.2.
A set P belonging to Y is said to be an $(X, Y)$ -uniformly (w.r.t $σ \in Σ$ ) attracting set for the family of processes ${U_{σ} (t, τ)}_{σ \in Σ}$ if, for an arbitrary fixed $τ \in R$ and $B \in B (X)$ $\begin{matrix} lim_{t \to + \infty} (sup_{σ \in Σ} dist (U_{σ} (t, τ) B, P)) = 0 . \end{matrix}$
Definition 2.3.
A closed set $A \subset Y$ is said to be an $(X, Y)$ -uniform (w.r.t $σ \in Σ$ ) attractor for the family of processes ${U_{σ} (t, τ)}_{σ \in Σ}$ if it is $(X, Y)$ -uniformly (w.r.t $σ \in Σ$ ) attracting and is contained in any closed $(X, Y)$ -uniformly (w.r.t $σ \in Σ$ ) attracting set $A^{'}$ of the family of processes ${U_{σ} (t, τ)}_{σ \in Σ}$ , i.e., $A \subset A^{'}$ .
Definition 2.4.
A family of processes ${U_{σ} (t, τ)}_{σ \in Σ}$ possessing a compact $(X, Y)$ -uniformly (w.r.t $σ \in Σ$ ) absorbing set is called $(X, Y)$ -uniformly compact. A family of processes ${U_{σ} (t, τ)}_{σ \in Σ}$ is called $(X, Y)$ -uniformly asymptotically compact if it possess a compact $(X, Y)$ -uniformly (w.r.t $σ \in Σ$ ) attracting set.
Definition 2.5.
The kernel $K$ of the process ${U (t, τ)}_{σ \in Σ}$ acting on X consists of all bounded complete trajectories of the process ${U (t, τ)}_{σ \in Σ}$ : $\begin{array}{l} K = {u (\cdot) | U (t, τ) u (τ) = u (t), dist (u (t), u (0)) ⩽ C_{u}, \forall t ⩾ τ, τ \in R} . \end{array}$ The set $K (s) = {u (s) | u (\cdot) \in K}$ is said to be kernel section at time $t = s$ , $s \in R$ .
Definition 2.6.
Let $L_{loc}^{2, w} (R; L^{2} (Ω))$ denote the space $L_{loc}^{2} (R; L^{2} (Ω))$ but under the local weak convergence topology. A sequence ${g_{n}}_{n = 1}^{\infty}$ is said to be weakly convergent to g (as $n \to \infty$ ) in $L_{loc}^{2} (R; L^{2} (Ω))$ if and only if $\begin{array}{l} \int_{t_{1}}^{t_{2}} \int_{Ω} v (x, s) (g_{n} (x, s) - g (x, s)) d x d s \to 0 (as n \to \infty) \end{array}$ for all $[t_{1}, t_{2}] \subset R$ and for every $v (x, s) \in L^{2} (t_{1}, t_{2}; L^{2} (Ω))$ .
Definition 2.7.
${U_{σ} (t, τ)}_{σ \in Σ}$ is said to be $(X \times Σ, Y)$ -weakly continuous if, for any fixed $t ⩾ τ$ , $τ \in R$ , the mapping $(u, σ) \to U_{σ} (t, τ) u$ is weakly continuous from $X \times Σ$ to Y.
Definition 2.8 ([21]).

A function $φ \in L_{loc}^{2} (R; X)$ is said to be normal if for any $ε > 0$ , there exists $η > 0$ such that $\begin{matrix} sup_{t \in R} \int_{t}^{t + η} {‖ φ (s) ‖}_{X}^{2} d s ⩽ ε . \end{matrix}$

Denote by $L_{n}^{2} (R; X)$ the set of all normal functions in $L_{loc}^{2} (R; X)$ .

Lemma 2.9 ([21]).

If $φ_{0} \in L_{n}^{2} (R; X)$ , then for any $τ \in R$ , $\begin{matrix} lim_{γ \to + \infty} sup_{t ⩾ τ} \int_{τ}^{t} e^{- γ (t - s)} {‖ φ (s) ‖}_{X}^{2} d s = 0 \end{matrix}$

uniformly w.r.t. $φ \in H_{w} (φ_{0})$ . Moreover, $L_{c}^{2} (R; X) \subset L_{n}^{2} (R; X) \subset L_{b}^{2} (R; X)$ .

Definition 2.10.
A family of processes ${U_{σ} (t, τ)}_{σ \in Σ}$ is said to be satisfied uniformly (w.r.t $σ \in Σ$ ) Condition $(C)$ if for any fixed $τ \in R$ , $B \in B (X)$ and $ε > 0$ , there exist $t_{0} = t_{0} (τ, B, ε) ⩾ τ$ and a finite dimensional subspace $X_{1} (\subset X)$ such that
$P (⋃_{σ \in Σ} ⋃_{t ⩾ t_{0}} U_{σ} (t, τ) B)$ is bounded;

$‖ (I - P) (⋃_{σ \in Σ} ⋃_{t ⩾ t_{0}} U_{σ} (t, τ) x) ‖ ⩽ ε$ , $\forall x \in B$ ,
where $P : X \to X_{1}$ is a bounded projector.
Definition 2.11.
A family of processes ${U_{σ} (t, τ)}_{σ \in Σ}$ is said to be uniformly (w.r.t $σ \in Σ$ ) ω-limit compact if for any $ε > 0$ and any $B \in B (X)$ , there exists a $t_{0} = t_{0} (τ, B, ε) ⩾ τ$ such that $\begin{array}{l} α (⋃_{σ \in Σ} ⋃_{s ⩾ t_{0}} U_{σ} (s, τ) B) ⩽ ε, \end{array}$ where $α (B)$ denotes the measure of noncompactness of B.
Lemma 2.12.
Suppose that ${U_{σ} (t, τ)}_{σ \in Σ}$ be a family of processes on Banach space X. If ${U_{σ} (t, τ)}_{σ \in Σ}$ satisfies the uniformly (w.r.t $σ \in Σ$ ) Condition $(C)$ , then ${U_{σ} (t, τ)}_{σ \in Σ}$ is uniformly (w.r.t $σ \in Σ$ ) ω-limit compact. Moreover, if X is a uniformly convex Banach space then the converse is true.
Lemma 2.13.
Let ${U_{σ} (t, τ)}_{σ \in Σ}$ be a family of processes on Banach space X and satisfy the translation identity ( 2.1 )–( 2.2 ). Then ${U_{σ} (t, τ)}_{σ \in Σ}$ has a uniform (w.r.t $σ \in Σ$ ) attractor in X if the following conditions hold:
${U_{σ} (t, τ)}_{σ \in Σ}$ has a bounded uniformly (w.r.t $σ \in Σ$ ) absorbing set B in X;

${U_{σ} (t, τ)}_{σ \in Σ}$ satisfies the uniformly (w.r.t $σ \in Σ$ ) Condition $(C)$ .

Lemma 2.14.
Let Σ be a weakly compact set and ${U_{σ} (t, τ)}_{σ \in Σ}$ be $(X \times Σ, Y)$ -weakly continuous with ${T (h)}_{h ⩾ 0}$ , which is a weakly continuous semigroup. If ${U_{σ} (t, τ)}_{σ \in Σ}$ , acting on Banach space X, satisfies the translation identity ( 2.1 )–( 2.2 ) and possesses a $(X, Y)$ -uniform (w.r.t $σ \in Σ$ ) attractor $A$ , which is compact in Y and attracts the bounded subset of X in the topology of Y, then the attractor can be decomposed as $\begin{array}{l} A = ⋃_{σ \in Σ} K_{σ} (0), \end{array}$ where $K_{σ}$ is the kernel of the family of processes ${U_{σ} (t, τ)}_{σ \in Σ}$ , $K_{σ} (0)$ is the kernel section at $t = 0$ .

2.2. Abstract results

In this subsection, we give the following abstract results, which are similar to those in [29,35] and used to verify the existence of the $(C_{L^{2} (Ω)}, C_{L^{p} (Ω)})$ -uniform attractors.

Lemma 2.15.
Let ${U_{σ} (t, τ)}_{σ \in Σ}$ be a family of processes on $C_{L^{p} (Ω)}$ ( $p ⩾ 1$ ) and have a bounded uniformly (w.r.t $σ \in Σ$ ) absorbing set in $C_{L^{p} (Ω)}$ . Then, for any $ε > 0$ , $τ \in R$ and any bounded subset $B \subset C_{L^{p} (Ω)}$ , there exist positive constants $T_{1} = T_{1} (B)$ and $M = M (ε, B)$ such that $\begin{array}{l} m (Ω (| U_{σ} (t + θ, τ) u (τ) | ⩾ M)) ⩽ ε for any u (τ) \in B, σ \in Σ, t - h ⩾ T_{1}, \end{array}$ where $m (Ω_{0})$ denotes the Lebesgue measure of $Ω_{0} \subset Ω$ and $Ω (| U_{σ} (t + θ, τ) u (τ) | ⩾ M) ≜ {x \in Ω; | u (t + θ) | ⩾ M}$ with $θ \in [- h, 0]$ .
Proof.
The process ${U_{σ} (t, τ)}_{σ \in Σ}$ has a bounded uniformly (w.r.t $σ \in Σ$ ) absorbing set in $C_{L^{p} (Ω)}$ , then there exists a constant $M_{0} > 0$ , such that for any $τ \in R$ and any bounded subset B of $C_{L^{p} (Ω)}$ , we can find a constant $T_{1} = T_{1} (B)$ such that $\begin{array}{l} max_{θ \in [- h, 0]} {‖ U_{σ} (t + θ, τ) u (τ) ‖}_{p}^{p} ⩽ M_{0} for any u (τ) \in B, σ \in Σ and t - h ⩾ T_{1} . \end{array}$

Therefore, $\begin{array}{l} M_{0} & ⩾ max_{θ \in [- h, 0]} \int_{Ω} {| U_{σ} (t + θ, τ) u (τ) |}^{p} d x \\ ⩾ max_{θ \in [- h, 0]} \int_{Ω (| U_{σ} (t + θ, τ) u (τ) | ⩾ M)} {| U_{σ} (t + θ, τ) u (τ) |}^{p} d x \\ ⩾ max_{θ \in [- h, 0]} \int_{Ω (| U_{σ} (t + θ, τ) u (τ) | ⩾ M)} M^{p} d x \\ = M^{p} \cdot m (Ω (| U_{σ} (t + θ, τ) u (τ) | ⩾ M)), \end{array}$ which implies that $m (Ω (| U_{σ} (t + θ, τ) u (τ) | ⩾ M)) ⩽ ε$ if we choose M large enough such that $M ⩾ {(\frac{M_{0}}{ε})}^{\frac{1}{p}}$ . □
Lemma 2.16.
Let B be a bounded subset of $C_{L^{p} (Ω)}$ ( $p > 2$ ). Then for any $ε > 0$ , the set B has a finite ε-net in $C_{L^{p} (Ω)}$ if there exists a positive constant $M = M (ε)$ which depends on ε, such that
B has a finite ${(3 M)}^{(2 - p) / 2} {(\frac{ε}{2})}^{p / 2}$ -net in $C_{L^{2} (Ω)}$ ;

$\begin{array}{l} (2.3) & {(\int_{Ω (u (t + θ) ⩾ M)} {| u (t + θ) |}^{p})}^{1 / p} < 2^{- (2 p + 2) / p} ε for any u (t) \in B, θ \in [- h, 0] . \end{array}$

Proof.
For any fixed $ε > 0$ , by the assumptions that B has a finite ${(3 M)}^{(2 - p) / 2} {(\frac{ε}{2})}^{p / 2}$ -net in $C_{L^{2} (Ω)}$ , then there exist $u_{1} (t), u_{2} (t), \dots, u_{k} (t) \in B$ , such that for any $u (t) \in B$ , we can find some $u_{i} (t)$ ( $1 ⩽ i ⩽ k$ ) satisfying $\begin{array}{l} {‖ u (t + θ) - u_{i} (t + θ) ‖}_{2}^{2} & ⩽ max_{θ \in [- h, 0]} {‖ u (t + θ) - u_{i} (t + θ) ‖}_{2}^{2} \\ = max_{θ \in [- h, 0]} {‖ u_{t} (θ) - u_{i t} (θ) ‖}_{2}^{2} \\ < {(3 M)}^{(2 - p)} {(\frac{ε}{2})}^{p} . \end{array}$

Then, we have $\begin{array}{l} {‖ u (t + θ) - u_{i} (t + θ) ‖}_{p}^{p} = & {‖ u_{t} (θ) - u_{i t} (θ) ‖}_{p}^{p} \\ = & \int_{Ω (| u (t + θ) - u_{i} (t + θ) | ⩾ 3 M)} {| u (t + θ) - u_{i} (t + θ) |}^{p} d x \\ (2.4) & + \int_{Ω (| u (t + θ) - u_{i} (t + θ) | ⩽ 3 M)} {| u (t + θ) - u_{i} (t + θ) |}^{p} d x, \end{array}$ and $\begin{array}{l} \int_{Ω (| u (t + θ) - u_{i} (t + θ) | ⩽ 3 M)} {| u (t + θ) - u_{i} (t + θ) |}^{p} d x \\ ⩽ {(3 M)}^{p - 2} \int_{Ω (| u (t + θ) - u_{i} (t + θ) | ⩽ 3 M)} {| u (t + θ) - u_{i} (t + θ) |}^{2} d x \\ ⩽ {(3 M)}^{p - 2} \cdot {(3 M)}^{(2 - p)} {(\frac{ε}{2})}^{p} \\ (2.5) & = {(\frac{ε}{2})}^{p} . \end{array}$

On the other hand, set $\begin{array}{l} Ω_{1} = Ω (| u (t + θ) | ⩾ \frac{3 M}{2}) \cap Ω (| u_{i} (t + θ) | ⩽ \frac{3 M}{2}), \\ (2.6) & Ω_{2} = Ω (| u (t + θ) | ⩽ \frac{3 M}{2}) \cap Ω (| u_{i} (t + θ) | ⩾ \frac{3 M}{2}), \\ Ω_{3} = Ω (| u (t + θ) | ⩾ \frac{3 M}{2}) \cap Ω (| u_{i} (t + θ) | ⩾ \frac{3 M}{2}), \end{array}$ then we have $\begin{array}{l} Ω (| u (t + θ) - u_{i} (t + θ) | ⩾ 3 M) \subset Ω_{1} \cup Ω_{2} \cup Ω_{3} . \end{array}$

From (2.6) we know that $| u (t + θ) - u_{i} (t + θ) | ⩽ 2 | u (t + θ) |$ in $Ω_{1}$ and $| u (t + θ) - u_{i} (t + θ) | ⩽ 2 | u_{i} (t + θ) |$ in $Ω_{2}$ , combining with (2.3), we have $\begin{array}{l} \int_{Ω (| u (t + θ) - u_{i} (t + θ) | ⩾ 3 M)} {| u (t + θ) - u_{i} (t + θ) |}^{p} d x \\ ⩽ \int_{Ω_{1}} {| u (t + θ) - u_{i} (t + θ) |}^{p} d x + \int_{Ω_{2}} {| u (t + θ) - u_{i} (t + θ) |}^{p} d x \\ + \int_{Ω_{3}} {| u (t + θ) - u_{i} (t + θ) |}^{p} d x \\ ⩽ 2^{p} \int_{Ω_{1}} {| u (t + θ) |}^{p} d x + 2^{p} \int_{Ω_{2}} {| u_{i} (t + θ) |}^{p} d x + 2^{p} \int_{Ω_{3}} {| u (t + θ) |}^{p} d x \\ + 2^{p} \int_{Ω_{3}} {| u_{i} (t + θ) |}^{p} d x \\ ⩽ 2^{p} (\int_{Ω (| u (t + θ) | ⩾ M)} {| u (t + θ) |}^{p} d x + \int_{Ω (| u_{i} (t + θ) | ⩾ M)} {| u_{i} (t + θ) |}^{p} d x) \\ + 2^{p} (\int_{Ω (| u (t + θ) | ⩾ M)} {| u (t + θ) |}^{p} d x + \int_{Ω (| u_{i} (t + θ) | ⩾ M)} {| u_{i} (t + θ) |}^{p} d x) \\ (2.7) & ⩽ 2^{p + 2} \cdot 2^{- (2 p + 2)} ε^{p} = {(\frac{ε}{2})}^{p} . \end{array}$

Substituting (2.5) and (2.7) into (2.4), we can deduce that $\begin{array}{l} max_{θ \in [- h, 0]} {‖ u (t + θ) - u_{i} (t + θ) ‖}_{p} < \frac{ε}{2} + \frac{ε}{2} = ε, \end{array}$ which means that the set B has a finite ε-net in $C_{L^{p} (Ω)}$ . □
Lemma 2.17.
Let B be a bounded subset of $C_{L^{p} (Ω)}$ ( $p ⩾ 1$ ). If B has a finite ε-net in $C_{L^{p} (Ω)}$ , then there exists a positive $M = M (ε, B)$ , such that for any $u (t) \in B$ , the following estimate holds $\begin{array}{l} max_{θ \in [- h, 0]} \int_{Ω (| u (t + θ) | ⩾ M)} {| u (t + θ) |}^{p} ⩽ 2^{p + 1} ε^{p} . \end{array}$
Proof.
Since B has a finite ε-net in $C_{L^{p} (Ω)}$ , then there exist $u_{1} (t), u_{2} (t), \dots, u_{k} (t) \in B$ , such that for any $u (t) \in B$ , we can find some $u_{i} (t)$ ( $1 ⩽ i ⩽ k$ ) satisfying $\begin{array}{l} (2.8) & max_{θ \in [- h, 0]} \int_{Ω} {| u (t + θ) - u_{i} (t + θ) |}^{p} ⩽ ε^{p} . \end{array}$

At the same time, for the fixed $ε > 0$ , there exists a $δ_{0} > 0$ , such that for each $u_{i} (t) \in B$ , $1 ⩽ i ⩽ k$ , we have $\begin{array}{l} (2.9) & max_{θ \in [- h, 0]} \int_{Ω_{0}} {| u_{i} (t + θ) |}^{p} d x ⩽ ε^{p}, \end{array}$ provided that $m (Ω_{0}) < δ_{0} (Ω_{0} \subset Ω)$ .

On the other hand, since B is bounded in $C_{L^{p} (Ω)}$ , for the $δ_{0} > 0$ given above, there exist $M > 0$ and $θ \in [- h, 0]$ , such that $m (Ω (| u (t + θ) | ⩾ M)) < δ_{0}$ holds for each $u (t) \in B$ .

Combining with (2.8) and (2.9), we immediately know that $\begin{array}{l} max_{θ \in [- h, 0]} \int_{Ω (| u (t + θ) | ⩾ M)} {| u (t + θ) |}^{p} d x \\ = max_{θ \in [- h, 0]} \int_{Ω (| u (t + θ) | ⩾ M)} {| u (t + θ) - u_{i} (t + θ) + u_{i} (t + θ) |}^{p} d x \\ ⩽ 2^{p} max_{θ \in [- h, 0]} \int_{Ω (| u (t + θ) | ⩾ M)} {| u (t + θ) - u_{i} (t + θ) |}^{p} d x \\ + 2^{p} max_{θ \in [- h, 0]} \int_{Ω (| u (t + θ) | ⩾ M)} {| u_{i} (t + θ) |}^{p} d x \\ ⩽ 2^{p + 1} ε^{p} . \end{array}$ □

Being similar to that in [29], we have the following conclusion, which is useful to verify the existence of the ( $C_{L^{2} (Ω)}$ , $C_{L^{p} (Ω)}$ )-uniform (w.r.t $σ \in Σ$ ) attractors. Theorem 2.18.
Let ${U_{σ} (t, τ)}_{σ \in Σ}$ be a family of processes on $C_{L^{2} (Ω)}$ and $C_{L^{p} (Ω)}$ ( $p > 2$ ). Suppose that ${U_{σ} (t, τ)}_{σ \in Σ}$ has a $(C_{L^{2} (Ω)}, C_{L^{2} (Ω)})$ -uniform (w.r.t $σ \in Σ$ ) attractor, then ${U_{σ} (t, τ)}_{σ \in Σ}$ has a $(C_{L^{2} (Ω)}, C_{L^{p} (Ω)})$ -uniform (w.r.t $σ \in Σ$ ) attractor provided that the following conditions hold:
${U_{σ} (t, τ)}_{σ \in Σ}$ has a $(C_{L^{2} (Ω)}, C_{L^{p} (Ω)})$ -bounded uniform (w.r.t $σ \in Σ$ ) absorbing set $B_{p}$ ;

for any $ε > 0$ , $τ \in R$ and bounded (with respect to $‖ \cdot ‖_{2}$ ) subset B, there exist constant $M = M (ε, B) > 0$ and $T_{2} = T_{2} (B, ε, τ)$ such that $\begin{array}{l} max_{θ \in [- h, 0]} \int_{Ω (| u (t + θ) | ⩾ M)} {| u (t + θ) |}^{p} < ε, for any u (τ) \in B, t - h ⩾ T_{2}, σ \in Σ . \end{array}$

Proof.
${U_{σ} (t, τ)}_{σ \in Σ}$ has a $(C_{L^{2} (Ω)}, C_{L^{2} (Ω)})$ -uniform (w.r.t $σ \in Σ$ ) attractor $A_{2}$ , then ${U_{σ} (t, τ)}_{σ \in Σ}$ has a $(C_{L^{2} (Ω)}, C_{L^{2} (Ω)})$ -uniform (w.r.t $σ \in Σ$ ) absorbing set, which denoted by $B_{2}$ . By condition $(1)$ and the embedding $C_{L^{p} (Ω)} ↪ C_{L^{2} (Ω)}$ ( $p > 2$ ), we know that $B_{p}$ is not only a $(C_{L^{2} (Ω)}, C_{L^{p} (Ω)})$ -uniform (w.r.t $σ \in Σ$ ) absorbing set, but also a $(C_{L^{2} (Ω)}, C_{L^{2} (Ω)})$ -uniform (w.r.t $σ \in Σ$ ) absorbing set.

Set $\begin{array}{l} (2.10) & A_{p} = ⋂_{t ⩾ τ} {\overline{⋃_{σ \in Σ} ⋃_{s ⩾ t} U_{σ} (s, τ) B_{p}}}^{C_{L^{p}}}, \forall τ \in R, \end{array}$ where ${\overline{A}}^{C_{L^{p}}}$ denotes the closure of A with respect to the $C_{L^{p}}$ -norm.

Now, we will verify the family of processes ${U_{σ} (t, τ)}_{σ \in Σ}$ is $(C_{L^{2} (Ω)}, C_{L^{p} (Ω)})$ -uniformly (w.r.t $σ \in Σ$ ) asymptotically compact. Denote the $(C_{L^{2} (Ω)}, C_{L^{p} (Ω)})$ -bounded uniform (w.r.t $σ \in Σ$ ) absorbing set by $B_{p}$ , then it is sufficient to prove that $\begin{matrix} for any x_{n} \in B_{p}, σ_{n} \in Σ, t_{n} \to + \infty, {U_{σ_{n}} (t_{n}, τ) x_{n}}_{n = 1}^{\infty} is precompact in C_{L^{p} (Ω)}, \end{matrix}$ which is equivalent to show that for any $ε > 0$ , ${U_{σ_{n}} (t_{n}, τ) x_{n}}_{n = 1}^{\infty}$ has a finite ε-net in $C_{L^{p} (Ω)}$ .

In fact, by the assumption that ${U_{σ} (t, τ)}_{σ \in Σ}$ has a $(C_{L^{2} (Ω)}, C_{L^{2} (Ω)})$ -uniform (w.r.t $σ \in Σ$ ) attractor, we know that there exists a $T_{3}$ , which depends on ε and M, such that ${U_{σ_{n}} (t_{n}, τ) x_{n} | t_{n} - h ⩾ T_{3}}$ has a finite ${(3 M)}^{(2 - p) / 2} {(\frac{ε}{2})}^{p / 2}$ -net in $C_{L^{2} (Ω)}$ . Let $T_{0} = max {T_{2}, T_{3}}$ , then from Lemma 2.16, we know that ${U_{σ_{n}} (t_{n}, τ) x_{n} | t_{n} - h ⩾ T_{0}}$ has a finite ε-net in $C_{L^{p} (Ω)}$ . Since $t_{n} \to + \infty$ , we obtain that ${U_{σ_{n}} (t_{n}, τ) x_{n}}_{n = 1}^{\infty}$ has a finite ε-net in $C_{L^{p} (Ω)}$ , too. By the arbitrariness of ε, we know that ${U_{σ_{n}} (t_{n}, τ) x_{n}}_{n = 1}^{\infty}$ is precompact in $C_{L^{p} (Ω)}$ .

Then, by the theory of dynamical systems (see, e.g., [1,10,16,26,30]), we know that ${U_{σ} (t, τ)}_{σ \in Σ}$ has a $(C_{L^{2} (Ω)}, C_{L^{p} (Ω)})$ -uniform (w.r.t $σ \in Σ$ ) attractor $A_{p}$ , and $A_{p}$ is compact in $C_{L^{p} (Ω)}$ and uniformly (w.r.t $σ \in Σ$ ) attracts every bounded subset of $C_{L^{2} (Ω)}$ with the $C_{L^{p} (Ω)}$ -norm. □
Remark 2.19.
From Theorem 2.18, we know that the $(C_{L^{2} (Ω)}, C_{L^{2} (Ω)})$ -uniform (w.r.t $σ \in Σ$ ) attractor $A_{2}$ coincides with the $(C_{L^{2} (Ω)}, C_{L^{p} (Ω)})$ -uniform (w.r.t $σ \in Σ$ ) attractor $A_{p}$ .

3. Uniform attractors for equation (1.1)

3.1. Existence and uniqueness results

In this subsection, we will give the well-posedness of solutions for equation (1.1). We first define the weak solutions, which is similar to that in [11], as follows.

Definition 3.1.
A weak solution of equation (1.1) is a function $u \in C ([τ - h, T]; L^{2} (Ω)) \cap L^{2} (τ, T; H_{0}^{1} (Ω)) \cap L^{p} (τ, T; L^{p} (Ω))$ for all $T > τ$ , with $u (t) = ϕ (t - τ)$ for all $t \in [τ - h, τ]$ and for all $φ \in H_{0}^{1} (Ω) \cap L^{p} (Ω)$ , it satisfies $\begin{array}{l} \frac{d}{d t} (u (t), φ) + (\nabla u (t), \nabla φ) = (f (u (t)), φ) + (g (u_{t}), φ) + (k (t), φ), in D^{'} (τ, + \infty) . \end{array}$

The following theorem gives the existence and uniqueness of solutions, which can be obtained by the Faedo–Galerkin method (see [11]). Here we only state the results:
Lemma 3.2.
Let f satisfy ( 1.2 )–( 1.3 ), $g (\cdot)$ subject to assumptions (I)–(II), $k (\cdot) \in L_{loc}^{2} (R; L^{2} (Ω))$ and $ϕ \in C_{L^{2} (Ω)}$ . Then for any $τ \in R$ and $T > τ$ , there exists a unique solution $u (\cdot) = u (\cdot; τ, ϕ)$ for equation ( 1.1 ), which satisfies $\begin{matrix} u \in C ([τ - h, T]; L^{2} (Ω)) \cap L^{2} (τ, T; H_{0}^{1} (Ω)) \cap L^{p} (τ, T; L^{p} (Ω)), \partial_{t} u \in L^{2} (τ, T; L^{2} (Ω)) . \end{matrix}$

Thus, we can define the family of processes ${U_{σ} (t, τ)}_{σ \in Σ}$ in $C_{L^{2} (Ω)}$ as follows: $\begin{array}{l} (3.1) & U_{σ} (t, τ) u (τ) = u (t), \forall t ⩾ τ, \end{array}$ and ${U_{σ} (t, τ)}_{σ \in Σ}$ is continuous in $C_{L^{2} (Ω)}$ .

From Lemma 3.2, we know that the family of processes ${U_{σ} (t, τ)}_{σ \in Σ}$ act in $C_{L^{2} (Ω)}$ and the time symbol is $σ (s) = Σ$ with $Σ = H_{w} (k)$ . We denote by $L_{loc}^{2, w} (R; L^{2} (Ω))$ the space $L_{loc}^{2} (R; L^{2} (Ω))$ endowed with a local weak convergence topology. Let $H_{w} (k)$ be the hull of k in $L_{loc}^{2, w} (R; L^{2} (Ω))$ , i.e., the closure of the set ${k (s + h^{'}) | h^{'} \in R}$ in $L_{loc}^{2, w} (R; L^{2} (Ω))$ and $k (x, s) \in L_{b}^{2} (R; L^{2} (Ω))$ .
Proposition 3.3 (See [10]).

If H is reflexive separable, $φ \in L_{b}^{2} (R; H)$ , then

for all $φ_{1} \in H_{ω} (φ)$ , $‖ φ_{1} ‖_{L_{b}^{2}}^{2} ⩽ ‖ φ ‖_{L_{b}^{2}}^{2}$ ;

the translation group $T (t)$ is weakly continuous on $H_{ω} (φ)$ ;

$T (t) H_{ω} (φ) = H_{ω} (φ)$ for $t ⩾ 0$ ;

$H_{ω} (φ)$ is weakly compact.

Due to Proposition 3.3, Σ is weakly compact and the translation semigroup ${T (h^{'}) | h^{'} ⩾ 0}$ satisfies that $T (h^{'}) Σ = Σ$ and is weakly continuous on Σ. Because of the uniqueness of the solutions, the following translation identity holds $\begin{array}{l} (3.2) & U_{σ} (t + h^{'}, τ + h^{'}) = U_{T (h^{'}) σ} (t, τ) for any σ \in Σ, t ⩾ τ, τ \in R, h^{'} ⩾ 0 . \end{array}$

Remark 3.4.
In the following, in this paper, we always assume that ${U_{σ} (t, τ)}_{σ \in Σ}$ is the family of processes defined by (3.1) and satisfies (3.2).

3.2. $(C_{L^{2} (Ω)}, C_{L^{2} (Ω)})$ -uniform attractors

The main purpose of this subsection is to prove the existence of the $(C_{L^{2} (Ω)}, C_{L^{2} (Ω)})$ -uniform (w.r.t $σ \in Σ$ ) attractors for ${U_{σ} (t, τ)}_{σ \in Σ}$ .

At first, we give the following estimates, which will be used to obtain the $(C_{L^{2} (Ω)}, C_{L^{2} (Ω)})$ -bounded uniform (w.r.t $σ \in Σ$ ) absorbing sets for ${U_{σ} (t, τ)}_{σ \in Σ}$ .

Lemma 3.5.
Let f satisfy ( 1.2 )–( 1.3 ), $g (\cdot)$ subject to assumptions (I)–(II), $k (\cdot) \in L_{b}^{2} (R, L^{2} (Ω))$ and $ϕ \in C_{L^{2} (Ω)}$ . Then the weak solutions $u (t)$ of equation ( 1.1 ) satisfies the following estimates: $\begin{array}{l} ‖ u_{t} ‖_{C_{L^{2} (Ω)}}^{2} ⩽ & e^{λ_{1} h - δ_{2} (t - τ)} ‖ ϕ ‖_{C_{L^{2} (Ω)}}^{2} + \frac{2 c_{0} | Ω | e^{λ_{1} h}}{δ_{2}} \\ (3.3) & + \frac{e^{λ_{1} h}}{δ_{2} (1 - e^{- δ_{2}})} sup_{t \in R} \int_{t}^{t + 1} {‖ k (s) ‖}_{2}^{2} d s, \forall t - h ⩾ τ, \end{array}$ where $δ_{2} = λ_{1} - L_{g} e^{\frac{λ_{1} h}{2}}$ , $λ_{1}$ is the first eigenvalue of $- Δ$ in $H_{0}^{1} (Ω)$ .
Proof.
Multiplying (1.1) by $u (t)$ and integrating over $x \in Ω$ , we arrive at $\begin{array}{l} \frac{1}{2} \frac{d}{d t} ‖ u ‖_{2}^{2} + ‖ \nabla u ‖_{2}^{2} = (f (u), u) + (g (u_{t}), u) + (k (t), u) . \end{array}$

Thanks to (1.3), assumption (II), the Hölder and Young inequalities, we have $\begin{array}{l} \frac{1}{2} \frac{d}{d t} ‖ u ‖_{2}^{2} + ‖ \nabla u ‖_{2}^{2} ⩽ & c_{0} | Ω | - c_{2} ‖ u ‖_{p}^{p} + {‖ g (u_{t}) ‖}_{2} ‖ u ‖_{2} + {‖ k (t) ‖}_{2} ‖ u ‖_{2} \\ ⩽ & c_{0} | Ω | - c_{2} ‖ u ‖_{p}^{p} + \frac{L_{g}^{2}}{2 δ_{1}} ‖ u_{t} ‖_{C_{L^{2} (Ω)}}^{2} + \frac{1}{2 δ_{2}} {‖ k (t) ‖}_{2}^{2} \\ + \frac{δ_{1} + δ_{2}}{2} ‖ u ‖_{2}^{2} . \end{array}$

Let $δ_{1} + δ_{2} = λ_{1}$ , where $λ_{1}$ is the first eigenvalue of $- Δ$ in $H_{0}^{1} (Ω)$ , we can get $\begin{array}{l} (3.4) & \frac{d}{d t} ‖ u ‖_{2}^{2} + ‖ \nabla u ‖_{2}^{2} + 2 c_{2} ‖ u ‖_{p}^{p} ⩽ 2 c_{0} | Ω | + \frac{L_{g}^{2}}{δ_{1}} ‖ u_{t} ‖_{C_{L^{2} (Ω)}}^{2} + \frac{1}{δ_{2}} {‖ k (t) ‖}_{2}^{2} . \end{array}$

Furthermore, $\begin{array}{l} (3.5) & \frac{d}{d t} ‖ u ‖_{2}^{2} + λ_{1} ‖ u ‖_{2}^{2} ⩽ 2 c_{0} | Ω | + \frac{L_{g}^{2}}{δ_{1}} ‖ u_{t} ‖_{C_{L^{2} (Ω)}}^{2} + \frac{1}{δ_{2}} {‖ k (t) ‖}_{2}^{2} . \end{array}$

Multiplying (3.5) by $e^{λ_{1} t}$ and integrating it in $[τ, t]$ , we obtain $\begin{array}{l} e^{λ_{1} t} {‖ u (t) ‖}_{2}^{2} ⩽ & e^{λ_{1} τ} {‖ u (τ) ‖}_{2}^{2} + 2 c_{0} | Ω | \int_{τ}^{t} e^{λ_{1} s} d s + \frac{L_{g}^{2}}{δ_{1}} \int_{τ}^{t} e^{λ_{1} s} ‖ u_{s} ‖_{C_{L^{2} (Ω)}}^{2} d s \\ + \frac{1}{δ_{2}} \int_{τ}^{t} e^{λ_{1} s} {‖ k (s) ‖}_{2}^{2} d s . \end{array}$

In particular, putting $t + θ$ instead of t with $θ \in [- h, 0]$ , we deduce that $\begin{array}{l} e^{λ_{1} t} ‖ u_{t} ‖_{C_{L^{2} (Ω)}}^{2} ⩽ & e^{λ_{1} (h + τ)} ‖ ϕ ‖_{C_{L^{2} (Ω)}}^{2} + 2 c_{0} | Ω | e^{λ_{1} h} \int_{τ}^{t} e^{λ_{1} s} d s \\ + \frac{L_{g}^{2} e^{λ_{1} h}}{δ_{1}} \int_{τ}^{t} e^{λ_{1} s} ‖ u_{s} ‖_{C_{L^{2} (Ω)}}^{2} d s + \frac{e^{λ_{1} h}}{δ_{2}} \int_{τ}^{t} e^{λ_{1} s} {‖ k (s) ‖}_{2}^{2} d s \end{array}$ for any $t - h ⩾ τ$ .

By the Gronwall lemma, it yields $\begin{array}{l} e^{λ_{1} t} ‖ u_{t} ‖_{C_{L^{2} (Ω)}}^{2} ⩽ & e^{λ_{1} (h + τ)} e^{\frac{L_{g}^{2} e^{λ_{1} h}}{δ_{1}} (t - τ)} ‖ ϕ ‖_{C_{L^{2} (Ω)}}^{2} \\ + 2 c_{0} | Ω | e^{λ_{1} h} e^{\frac{L_{g}^{2} e^{λ_{1} h}}{δ_{1}} t} \int_{τ}^{t} e^{(λ_{1} - \frac{L_{g}^{2} e^{λ_{1} h}}{δ_{1}}) s} d s \\ + \frac{e^{λ_{1} h}}{δ_{2}} e^{\frac{L_{g}^{2} e^{λ_{1} h}}{δ_{1}} t} \int_{τ}^{t} e^{(λ_{1} - \frac{L_{g}^{2} e^{λ_{1} h}}{δ_{1}}) s} {‖ k (s) ‖}_{2}^{2} d s, \forall t - h ⩾ τ . \end{array}$

Let $δ_{1} = L_{g} e^{\frac{λ_{1} h}{2}}$ , then $δ_{2} = λ_{1} - L_{g} e^{\frac{λ_{1} h}{2}}$ and $\begin{array}{l} ‖ u_{t} ‖_{C_{L^{2} (Ω)}}^{2} ⩽ & e^{λ_{1} h} e^{- δ_{2} (t - τ)} ‖ ϕ ‖_{C_{L^{2} (Ω)}}^{2} + 2 c_{0} | Ω | e^{λ_{1} h} e^{- δ_{2} t} \int_{τ}^{t} e^{δ_{2} s} d s \\ (3.6) & + \frac{e^{λ_{1} h}}{δ_{2}} e^{- δ_{2} t} \int_{τ}^{t} e^{δ_{2} s} {‖ k (s) ‖}_{2}^{2} d s, \forall t - h ⩾ τ . \end{array}$

Moreover, $\begin{array}{l} e^{- δ_{2} t} \int_{τ}^{t} e^{δ_{2} s} {‖ k (s) ‖}_{2}^{2} d s \\ = e^{- δ_{2} t} (\int_{t - 1}^{t} e^{δ_{2} s} {‖ k (s) ‖}_{2}^{2} d s + \int_{t - 2}^{t - 1} e^{δ_{2} s} {‖ k (s) ‖}_{2}^{2} d s + \dots) \\ = e^{- δ_{2} t} (\int_{0}^{1} e^{δ_{2} (s + t - 1)} {‖ k (s + t - 1) ‖}_{2}^{2} d s + \int_{0}^{1} e^{δ_{2} (s + t - 2)} {‖ k (s + t - 2) ‖}_{2}^{2} d s + \dots) \\ ⩽ (e^{- δ_{2}} + e^{- 2 δ_{2}} + \dots) sup_{t \in R} \int_{0}^{1} e^{δ_{2} s} {‖ k (s + t) ‖}_{2}^{2} d s \\ ⩽ \frac{e^{- δ_{2}}}{1 - e^{- δ_{2}}} sup_{t \in R} \int_{0}^{1} e^{δ_{2} s} {‖ k (s + t) ‖}_{2}^{2} d s \\ = \frac{1}{1 - e^{- δ_{2}}} sup_{t \in R} \int_{0}^{1} {‖ k (s + t) ‖}_{2}^{2} d s \\ (3.7) & = \frac{1}{1 - e^{- δ_{2}}} sup_{t \in R} \int_{t}^{t + 1} {‖ k (s) ‖}_{2}^{2} d s . \end{array}$

Combining with (3.6) and (3.7), we obtain that $\begin{array}{l} ‖ u_{t} ‖_{C_{L^{2} (Ω)}}^{2} ⩽ e^{λ_{1} h - δ_{2} (t - τ)} ‖ ϕ ‖_{C_{L^{2} (Ω)}}^{2} + \frac{2 c_{0} | Ω | e^{λ_{1} h}}{δ_{2}} + \frac{e^{λ_{1} h}}{δ_{2} (1 - e^{- δ_{2}})} sup_{t \in R} \int_{t}^{t + 1} {‖ k (s) ‖}_{2}^{2} d s \end{array}$ for any $t - h ⩾ τ$ . □

As a direct corollary of Lemma 3.5, we immediately obtain the bounded uniform (w.r.t $σ \in Σ$ ) absorbing set in $C_{L^{2} (Ω)}$ for ${U_{σ} (t, τ)}_{σ \in Σ}$ .
Corollary 3.6.
Let f satisfy ( 1.2 )–( 1.3 ), $g (\cdot)$ subject to assumptions (I)–(II), $k (\cdot) \in L_{b}^{2} (R, L^{2} (Ω))$ and $ϕ \in C_{L^{2} (Ω)}$ . Then ${U_{σ} (t, τ)}_{σ \in Σ}$ has a $(C_{L^{2} (Ω)}, C_{L^{2} (Ω)})$ -bounded uniformly (w.r.t $σ \in Σ$ ) absorbing set, that is, there is a constant $ρ_{0} > 0$ such that for any bounded (with respect to $‖ \cdot ‖_{2}$ ) subset B, there exists a $T_{4} = T_{4} (‖ B ‖_{2})$ such that $\begin{array}{l} (3.8) & ‖ u_{t} ‖_{C_{L^{2} (Ω)}}^{2} ⩽ ρ_{0}^{2}, \forall t - h ⩾ T_{4}, \end{array}$ where $ρ_{0}^{2} = 1 + \frac{2 c_{0} | Ω | e^{λ_{1} h}}{δ_{2}} + \frac{e^{λ_{1} h}}{δ_{2} (1 - e^{- δ_{2}})} {sup}_{t \in R} \int_{t}^{t + 1} ‖ k (s) ‖_{2}^{2} d s$ , $T_{4} = τ + \frac{1}{δ_{2}} (λ_{1} h + 2 ln ‖ ϕ ‖_{C_{L^{2} (Ω)}})$ .
Proof.
It is a immediate consequence of (3.3) in Lemma 3.5, by taking $δ_{2} = λ_{1} - L_{g} e^{\frac{λ_{1} h}{2}} > 0$ and $T_{4} = τ + \frac{1}{δ_{2}} (λ_{1} h + 2 ln ‖ ϕ ‖_{C_{L^{2} (Ω)}})$ . □

In the following, we obtain the existence of the $(C_{L^{2} (Ω)}, C_{L^{2} (Ω)})$ -uniform (w.r.t $σ \in Σ$ ) attractors for ${U_{σ} (t, τ)}_{σ \in Σ}$ . Theorem 3.7.
Let f satisfy ( 1.2 )–( 1.3 ), $g (\cdot)$ subject to assumptions (I)–(II), $k (\cdot) \in L_{b}^{2} (R, L^{2} (Ω))$ and $ϕ \in C_{L^{2} (Ω)}$ . Then ${U_{σ} (t, τ)}_{σ \in Σ}$ has a $(C_{L^{2} (Ω)}, C_{L^{2} (Ω)})$ -uniform (w.r.t $σ \in Σ$ ) attractor, which is compact in $C_{L^{2} (Ω)}$ and uniformly (w.r.t $σ \in Σ$ ) attracts every bounded subset of $C_{L^{2} (Ω)}$ with the $C_{L^{2} (Ω)}$ -norm. Moreover, the attractor can be decomposed as $\begin{array}{l} (3.9) & A = ⋃_{σ \in Σ} K_{σ} (0), \end{array}$ where $K_{σ}$ is the kernel of the family of processes ${U_{σ} (t, τ)}_{σ \in Σ}$ , $K_{σ} (0)$ is the kernel section at time $t = 0$ .
Proof.
By Lemma 3.5 and Corollary 3.6, we know that there is a bounded uniformly (w.r.t $σ \in Σ$ ) absorbing set in $C_{L^{2} (Ω)}$ for ${U_{σ} (t, τ)}_{σ \in Σ}$ . Moreover, from Lemma 3.2 we know that $u \in L^{2} (t - h, t; H_{0}^{1} (Ω))$ and $\partial_{t} u \in L^{2} (t - h, t; L^{2} (Ω))$ for sufficiently large t. Then there exists a subsequence that converges strongly in $C_{L^{2} (Ω)}$ (see [20] for details). So the uniform (w.r.t $σ \in Σ$ ) absorbing set in $C_{L^{2} (Ω)}$ for ${U_{σ} (t, τ)}_{σ \in Σ}$ indeed is the uniform (w.r.t $σ \in Σ$ ) attractor in $C_{L^{2} (Ω)}$ for ${U_{σ} (t, τ)}_{σ \in Σ}$ .

Thanks to Proposition 3.3, we know that the symbol space Σ is weak continuity. Then combining with Lemma 2.14, we immediately obtain (3.9). □

3.3. $(C_{L^{2} (Ω)}, C_{L^{p} (Ω)})$ -uniform attractors

In this subsection, we will verify the existence of the $(C_{L^{2} (Ω)}), C_{L^{p} (Ω)}))$ -uniform (w.r.t $σ \in Σ$ ) attractors for ${U_{σ} (t, τ)}_{σ \in Σ}$ .

At first, we need the following lemma, which will be used to obtain the bounded uniform (w.r.t $σ \in Σ$ ) absorbing sets in $C_{L^{p} (Ω)}$ and $C_{H_{0}^{1} (Ω)}$ for ${U_{σ} (t, τ)}_{σ \in Σ}$ .

Lemma 3.8.
Let f satisfy ( 1.2 )–( 1.3 ), $g (\cdot)$ subject to assumptions (I)–(II), $k (\cdot) \in L_{b}^{2} (R, L^{2} (Ω))$ and $ϕ \in C_{L^{2} (Ω)}$ . Then ${U_{σ} (t, τ)}_{σ \in Σ}$ has a $(C_{L^{2} (Ω)}, C_{L^{p} (Ω)})$ and a $(C_{L^{2} (Ω)}, C_{H_{0}^{1} (Ω)})$ -bounded uniform (w.r.t $σ \in Σ$ ) absorbing set, that is, there are positive constants $ρ_{1}$ and $ρ_{2}$ such that for any bounded (with respect to $‖ \cdot ‖_{2}$ ) subset B, there exists a $T_{4} = T_{4} (B)$ such that $\begin{array}{l} ‖ \nabla u_{t} ‖_{C_{L^{2} (Ω)}}^{2} ⩽ ρ_{1}^{2} and ‖ u_{t} ‖_{C_{L^{p} (Ω)}}^{p} ⩽ ρ_{2}^{p}, \forall t - h ⩾ T_{4}, \end{array}$ where $ρ_{1}^{2} = 2 c_{0} C | Ω | + C (1 + \frac{L_{g}^{2}}{δ_{1}}) ρ_{0}^{2} + 2 L_{g}^{2} ρ_{0}^{2} + (2 + \frac{C}{δ_{2}}) {sup}_{t \in R} \int_{t}^{t + 1} ‖ k (s) ‖_{2}^{2} d s$ and $ρ_{2}^{p} = \frac{1}{2 \tilde{c_{2}}} (ρ_{1}^{2} + 2 \tilde{c_{0}} | Ω |)$ , $T_{4} = T_{4} (B)$ is given by Corollary 3.6 .
Proof.
From (3.4) in Lemma 3.5, we have $\begin{array}{l} \frac{d}{d t} ‖ u ‖_{2}^{2} + (‖ \nabla u ‖_{2}^{2} + 2 c_{2} ‖ u ‖_{p}^{p}) ⩽ 2 c_{0} | Ω | + \frac{L_{g}^{2}}{δ_{1}} ‖ u_{t} ‖_{C_{L^{2} (Ω)}}^{2} + \frac{1}{δ_{2}} {‖ k (t) ‖}_{2}^{2} . \end{array}$

Integrating the inequality above from t to $t + 1$ , we have $\begin{array}{l} \int_{t}^{t + 1} (‖ \nabla u ‖_{2}^{2} + 2 c_{2} ‖ u ‖_{p}^{p}) d s \\ ⩽ 2 c_{0} | Ω | + {‖ u (t) ‖}_{2}^{2} + \frac{L_{g}^{2}}{δ_{1}} \int_{t}^{t + 1} ‖ u_{s} ‖_{C_{L^{2} (Ω)}}^{2} d s + \frac{1}{δ_{2}} sup_{t \in R} \int_{t}^{t + 1} {‖ k (s) ‖}_{2}^{2} d s . \end{array}$

Combining with (3.8) in Corollary 3.6, we obtain that $\begin{array}{l} \int_{t}^{t + 1} (‖ \nabla u ‖_{2}^{2} + 2 c_{2} ‖ u ‖_{p}^{p}) d s ⩽ 2 c_{0} | Ω | + (1 + \frac{L_{g}^{2}}{δ_{1}}) ρ_{0}^{2} + \frac{1}{δ_{2}} sup_{t \in R} \int_{t}^{t + 1} {‖ k (s) ‖}_{2}^{2} d s, \end{array}$ for all $t - h ⩾ T_{4}$ , where $T_{4}$ comes from Corollary 3.6.

Thanks to (1.4), we have $\begin{array}{l} (3.10) & \int_{t}^{t + 1} (‖ \nabla u ‖_{2}^{2} - 2 (F (u), 1)) d s ⩽ & 2 c_{0} C | Ω | + C (1 + \frac{L_{g}^{2}}{δ_{1}}) ρ_{0}^{2} + \frac{C}{δ_{2}} sup_{t \in R} \int_{t}^{t + 1} {‖ k (s) ‖}_{2}^{2} d s, \end{array}$ for all $t - h ⩾ T_{4}$ .

On the other hand, multiplying (1.1) by $\partial_{t} u$ and integrating over Ω, we obtain $\begin{array}{l} ‖ \partial_{t} u ‖_{2}^{2} + \frac{1}{2} \frac{d}{d t} ‖ \nabla u ‖_{2}^{2} = (f (u), \partial_{t} u) + (g (u_{t}), \partial_{t} u) + (k (t), \partial_{t} u) . \end{array}$

Furthermore, by (1.4), assumption (II), the Hölder and Young inequalities, we have $\begin{array}{l} \frac{d}{d t} (‖ \nabla u ‖_{2}^{2} - 2 (F (u), 1)) & ⩽ 2 L_{g}^{2} ‖ u_{t} ‖_{C_{L^{2} (Ω)}}^{2} + 2 ‖ (k (t) ‖_{2}^{2} \\ (3.11) & ⩽ 2 L_{g}^{2} ρ_{0}^{2} + 2 ‖ (k (t) ‖_{2}^{2}, \forall t - h ⩾ T_{4}, \end{array}$ where $T_{4}$ and $ρ_{0}^{2}$ come from Corollary 3.6.

Combining with (3.10) and (3.11), by the uniform Gronwall lemma, we obtain that $\begin{array}{l} ‖ \nabla u ‖_{2}^{2} - 2 (F (u), 1) ⩽ ρ_{1}^{2} for all t - h ⩾ T_{4}, \end{array}$ where $ρ_{1}^{2} = 2 c_{0} C | Ω | + C (1 + \frac{L_{g}^{2}}{δ_{1}}) ρ_{0}^{2} + 2 L_{g}^{2} ρ_{0}^{2} + (2 + \frac{C}{δ_{2}}) {sup}_{t \in R} \int_{t}^{t + 1} ‖ k (s) ‖_{2}^{2} d s$ .

By (1.4), we know that $\begin{array}{l} {‖ \nabla u (t) ‖}_{2}^{2} ⩽ ρ_{1}^{2} and {‖ u (t) ‖}_{p}^{p} ⩽ ρ_{2}^{p} for all t - h ⩾ T_{4}, \end{array}$ where $ρ_{2}^{p} = \frac{1}{2 \tilde{c_{2}}} (ρ_{1}^{2} + 2 \tilde{c_{0}} | Ω |)$ .

In particular, putting $t + θ$ instead of t with $θ \in [- h, 0]$ , we have $\begin{array}{l} ‖ \nabla u_{t} ‖_{C_{L^{2} (Ω)}}^{2} ⩽ ρ_{1}^{2} and ‖ u_{t} ‖_{C_{L^{p} (Ω)}}^{p} ⩽ ρ_{2}^{p} for all t - h ⩾ T_{4}, \end{array}$ where $ρ_{2}^{p} = \frac{1}{2 \tilde{c_{2}}} (ρ_{1}^{2} + 2 \tilde{c_{0}} | Ω |)$ . □

Moreover, we give the asymptotic a priori estimates for ${U_{σ} (t, τ)}_{σ \in Σ}$ with respect to $C_{L^{p}}$ -norm, which plays a crucial role in the proof of the existence of the $(C_{L^{2} (Ω)}, C_{L^{p} (Ω)})$ -uniform (w.r.t $σ \in Σ$ ) attractors.
Theorem 3.9.
Let f satisfy ( 1.2 )–( 1.3 ), $g (\cdot)$ subject to assumptions (I)–(II), $k (\cdot) \in L_{b}^{2} (R, L^{2} (Ω))$ and $ϕ \in C_{L^{2} (Ω)}$ . Assume further $k (\cdot) \in L_{n}^{2} (R, L^{2} (Ω))$ , then for any $ε > 0$ , there exist $M = M (ε)$ and $t_{0} (⩾ T_{4} + h)$ such that $\begin{array}{l} (3.12) & max_{θ \in [- h, 0]} \int_{Ω (| u (t + θ) | ⩾ M)} {| u (t + θ) |}^{p} d x & ⩽ C ε, for any t - h ⩾ t_{0}, \end{array}$ where the constant C is independent of M, $t_{0}$ and ε, $T_{4}$ comes from Corollary 3.6 .
Proof.
Multiplying (1.1) by $| {(u - M_{1})}_{+} |^{p - 2} {(u - M_{1})}_{+}$ and integrating over Ω, we obtain that $\begin{array}{l} \frac{1}{p} \frac{d}{d t} {‖ {(u - M_{1})}_{+} ‖}_{p}^{p} + (p - 1) \int_{Ω} {| \nabla {(u - M_{1})}_{+} |}^{2} {| {(u - M_{1})}_{+} |}^{p - 2} d x \\ = \int_{Ω} f (u) {| {(u - M_{1})}_{+} |}^{p - 1} d x + \int_{Ω} g (u_{t}) {| {(u - M_{1})}_{+} |}^{p - 1} d x \\ (3.13) & + \int_{Ω} k (t) {| {(u - M_{1})}_{+} |}^{p - 1} d x, \end{array}$ where ${(u - M_{1})}_{+}$ denotes the positive part of $u - M_{1}$ , that is $\begin{array}{l} {(u - M_{1})}_{+} = \{\begin{matrix} u - M_{1}, & u ⩾ M_{1}, \\ 0, & u < M_{1} . \end{matrix} \end{array}$

In view of (1.3), let $M_{1} = M_{1} (c_{0}, c_{2}, p)$ large enough, we have $\begin{array}{l} (3.14) & f (u) ⩽ - \frac{c_{2}}{2} | u |^{p - 1} for u ⩾ M_{1} . \end{array}$ In consequence, $\begin{array}{l} f (u) {| {(u - M_{1})}_{+} |}^{p - 1} & ⩽ - \frac{c_{2}}{2} | u |^{p - 1} {| {(u - M_{1})}_{+} |}^{p - 1} \\ ⩽ - \frac{c_{2}}{4} | u |^{p - 1} {| {(u - M_{1})}_{+} |}^{p - 1} - \frac{c_{2}}{4} {| {(u - M_{1})}_{+} |}^{2 p - 2} \\ (3.15) & ⩽ - \frac{c_{2}}{4} M_{1}^{p - 2} {| {(u - M_{1})}_{+} |}^{p} - \frac{c_{2}}{4} {| {(u - M_{1})}_{+} |}^{2 p - 2} . \end{array}$

Moreover, by assumption (II), the Hölder and Young inequalities, we have $\begin{array}{l} (3.16) & \int_{Ω} g (u_{t}) {(u - M_{1})}_{+}^{p - 1} d x ⩽ \frac{2 L_{g}^{2}}{c_{2}} ‖ u_{t} ‖_{C_{L^{2}}}^{2} + \frac{c_{2}}{8} {‖ {(u - M_{1})}_{+} ‖}_{2 p - 2}^{2 p - 2}, \end{array}$ and $\begin{array}{l} (3.17) & \int_{Ω} k (t) {(u - M_{1})}_{+}^{p - 1} d x ⩽ \frac{2}{c_{2}} {‖ k (t) ‖}_{2}^{2} + \frac{c_{2}}{8} {‖ {(u - M_{1})}_{+} ‖}_{2 p - 2}^{2 p - 2} . \end{array}$

Substituting (3.14)–(3.17) into (3.13), we obtain that $\begin{array}{l} \frac{1}{p} \frac{d}{d t} {‖ {(u - M_{1})}_{+} ‖}_{p}^{p} + \frac{c_{2}}{4} | M_{1} |^{p - 2} \int_{Ω} {(u - M_{1})}_{+}^{p} d x ⩽ \frac{2 L_{g}^{2}}{c_{2}} ‖ u_{t} ‖_{C_{L^{2}}}^{2} + \frac{2}{c_{2}} {‖ k (t) ‖}_{2}^{2} . \end{array}$

Furthermore, $\begin{array}{l} (3.18) & \frac{d}{d t} {‖ {(u - M_{1})}_{+} ‖}_{p}^{p} + \frac{c_{2} p}{4} M_{1}^{p - 2} \int_{Ω} {(u - M_{1})}_{+}^{p} d x ⩽ \frac{2 p L_{g}^{2}}{c_{2}} ‖ u_{t} ‖_{C_{L^{2}}}^{2} + \frac{2 p}{c_{2}} {‖ k (t) ‖}_{2}^{2} . \end{array}$

On the other hand, taking $| {(u + M_{1})}_{-} |^{p - 2} {(u + M_{1})}_{-}$ instead of $| {(u - M_{1})}_{+} |^{p - 2} {(u - M_{1})}_{+}$ in the preceding proof, we deduce similarly that $\begin{array}{l} (3.19) & \frac{d}{d t} {‖ {(u + M_{1})}_{-} ‖}_{p}^{p} + \frac{c_{2} p}{4} M_{1}^{p - 2} \int_{Ω} {(u + M_{1})}_{-}^{p} d x ⩽ \frac{2 p L_{g}^{2}}{c_{2}} ‖ u_{t} ‖_{C_{L^{2}}}^{2} + \frac{2 p}{c_{2}} {‖ k (t) ‖}_{2}^{2}, \end{array}$ where ${(u + M_{1})}_{-}$ denotes the negative part of $u + M_{1}$ , that is $\begin{array}{l} {(u + M_{1})}_{-} = \{\begin{matrix} u + M_{1}, & u ⩽ - M_{1}, \\ 0, & u > - M_{1} . \end{matrix} \end{array}$

Combining with (3.18) and (3.19), we obtain that $\begin{array}{l} \frac{d}{d t} {‖ {(| u (t) | - M_{1})}_{+} ‖}_{p}^{p} + \frac{c_{2} p}{4} M_{1}^{p - 2} \int_{Ω} {(| u (t) | - M_{1})}_{+}^{p} d x \\ (3.20) & ⩽ \frac{2 p L_{g}^{2}}{c_{2}} ‖ u_{t} ‖_{C_{L^{2}}}^{2} + \frac{2 p}{c_{2}} {‖ k (t) ‖}_{2}^{2} . \end{array}$

Setting $α_{1} = \frac{c_{2} p}{4} M_{1}^{p - 2}$ , $α_{2} = \frac{2 p L_{g}^{2}}{c_{2}}$ , $α_{3} = \frac{2 p}{c_{2}}$ and multiplying (3.20) by $e^{α_{1} t}$ , then applying the Gronwall lemma to (3.20) and integrating it from $t_{0}$ to t, we deduce that $\begin{array}{l} \int_{Ω (| u | ⩾ M_{1})} {| {(| u (t) | - M_{1})}_{+} |}^{p} d x \\ ⩽ e^{- α_{1} (t - t_{0})} \int_{Ω (| u | ⩾ M_{1})} {| {(| u (t_{0}) | - M_{1})}_{+} |}^{p} d x + α_{2} e^{- α_{1} t} \int_{t_{0}}^{t} e^{α_{1} s} ‖ u_{s} ‖_{C_{L^{2}}}^{2} d s \\ + α_{3} e^{- α_{1} t} \int_{t_{0}}^{t} e^{α_{1} s} \int_{Ω} {| k (x, s) |}^{2} d x d s \\ ⩽ e^{- α_{1} (t - t_{0})} \int_{Ω (| u | ⩾ M_{1})} {| u (t_{0}) |}^{p} d x + α_{2} e^{- α_{1} t} \int_{t_{0}}^{t} e^{α_{1} s} ‖ u_{s} ‖_{C_{L^{2}}}^{2} d s \\ + α_{3} e^{- α_{1} t} \int_{t_{0}}^{t} e^{α_{1} s} {‖ k (s) ‖}_{2}^{2} d s \\ (3.21) & = I_{1} + I_{2} + I_{3}, \end{array}$ where $t_{0} - h ⩾ T_{4}$ (see Corollary 3.6 and Lemma 3.8) and $u (t_{0}) \in B_{p}$ .

In the following, we will estimate each term on the right hand of (3.21).

Firstly, in view of (3.8) in Corollary 3.6, we immediately get that $\begin{array}{l} (3.22) & I_{2} = α_{2} e^{- α_{1} t} \int_{t_{0}}^{t} e^{α_{1} s} ‖ u_{s} ‖_{C_{L^{2}}}^{2} d s ⩽ α_{2} ρ_{0}^{2} e^{- α_{1} t} \int_{t_{0}}^{t} e^{α_{1} s} d s ⩽ \frac{α_{2}}{α_{1}} ρ_{0}^{2}, \end{array}$ where $ρ_{0}^{2} = 1 + \frac{2 c_{0} | Ω | e^{λ_{1} h}}{δ_{2}} + \frac{e^{λ_{1} h}}{δ_{2} (1 - e^{- δ_{2}})} {sup}_{t \in R} \int_{t}^{t + 1} ‖ k (s) ‖_{2}^{2} d s$ comes from (3.8).

Then, for any $ε > 0$ , noticing that $α_{1} = \frac{c_{2} p}{4} M_{1}^{p - 2}$ , so we can take $M_{1}$ large enough such that $\begin{array}{l} (3.23) & I_{2} = α_{2} e^{- α_{1} t} \int_{t_{0}}^{t} e^{α_{1} s} ‖ u_{s} ‖_{C_{L^{2}}}^{2} d s < \frac{ε}{3} . \end{array}$

Secondly, for the given $ε > 0$ and $α_{1}$ in (3.22), in view of Lemma 3.8, let $t_{1} = t_{0} + \frac{1}{α_{1}} ln \frac{3 ρ_{2}^{p}}{ε}$ , we can deduce that $\begin{array}{l} (3.24) & I_{1} = e^{- α_{1} (t - t_{0})} \int_{Ω (| u | ⩾ M_{1})} {| u (t_{0}) |}^{p} d x < \frac{ε}{3} for all t ⩾ t_{1} . \end{array}$

Finally, for the given $ε > 0$ , in view of Lemma 2.9, we can take $M_{1}$ large enough such that $\begin{array}{l} (3.25) & I_{3} = α_{3} e^{- α_{1} t} \int_{t_{0}}^{t} e^{α_{1} s} {‖ k (s) ‖}_{2}^{2} d s < \frac{ε}{3} . \end{array}$

Combining with (3.21)–(3.25), we can deduce that $\begin{array}{l} (3.26) & \int_{Ω (| u (t) | ⩾ M_{1})} {| {(| u (t) | - M_{1})}_{+} |}^{p} d x < ε . \end{array}$

Therefore, $\begin{array}{l} \int_{Ω (| u (t) | ⩾ 2 M_{1})} {| u (t) |}^{p} d x & = \int_{Ω (| u (t) | ⩾ 2 M_{1})} {((| u (t) | - M_{1}) + M_{1})}^{p} d x \\ ⩽ 2^{p} (\int_{Ω (| u (t) | ⩾ 2 M_{1})} {(| u (t) | - M_{1})}^{p} d x + \int_{Ω (| u (t) | ⩾ 2 M_{1})} M_{1}^{p}) \\ ⩽ 2^{p + 1} \int_{Ω (| u (t) | ⩾ 2 M_{1})} {(| u (t) | - M_{1})}^{p} d x \\ (3.27) & ⩽ 2^{p + 1} ε \end{array}$ as $| u (t) | - M_{1} ⩾ M_{1}$ for $| u (t) | ⩾ 2 M_{1}$ .

Setting $M = 2 M_{1}$ , $C = 2^{p + 1}$ and putting $t + θ$ instead of t with $θ \in [- h, 0]$ , we can deduce that $\begin{array}{l} max_{θ \in [- h, 0]} \int_{Ω (| u (t + θ) | ⩾ M)} {| u (t + θ) |}^{p} d x & ⩽ C ε for any t - h ⩾ t_{0} (⩾ T_{4} + h), \end{array}$ which complete the proof. □

In the following, we obtain the existence and the structure of the $(C_{L^{2} (Ω)}, C_{L^{p} (Ω)})$ -uniform (w.r.t $σ \in Σ$ ) attractors for ${U_{σ} (t, τ)}_{σ \in Σ}$ . Theorem 3.10.
Let f satisfy ( 1.2 )–( 1.3 ), $g (\cdot)$ subject to assumptions (I)–(II), $k (\cdot) \in L_{b}^{2} (R, L^{2} (Ω))$ and $ϕ \in C_{L^{2} (Ω)}$ . Assume further $k (\cdot) \in L_{n}^{2} (R, L^{2} (Ω))$ , then ${U_{σ} (t, τ)}_{σ \in Σ}$ has a $(C_{L^{2} (Ω)}, C_{L^{p} (Ω)})$ -uniform (w.r.t $σ \in Σ$ ) attractor, which is compact in $C_{L^{p} (Ω)}$ and uniformly (w.r.t $σ \in Σ$ ) attracts every bounded subset of $C_{L^{2} (Ω)}$ with the $C_{L^{p} (Ω)}$ -norm. Moreover, the attractor can be decomposed as $\begin{array}{l} (3.28) & A = ⋃_{σ \in Σ} K_{σ} (0), \end{array}$ where $K_{σ}$ is the kernel of the family of processes ${U_{σ} (t, τ)}_{σ \in Σ}$ , $K_{σ} (0)$ is the kernel section at time $t = 0$ .
Proof.
Combining with Lemma 3.8 and (3.12) in Theorem 3.9, we know that the conditions in Theorem 2.18 are all satisfied and immediately obtain the existence of the $(C_{L^{2} (Ω)}, C_{L^{p} (Ω)})$ -uniform (w.r.t $σ \in Σ$ ) attractors for ${U_{σ} (t, τ)}_{σ \in Σ}$ .

Moreover, thanks to Remark 2.19 and Theorem 3.7, we obtain (3.28) and complete the proof. □

3.4. $(C_{L^{2} (Ω)}, C_{H_{0}^{1} (Ω)})$ -uniform attractors

In this subsection, we will prove the existence of the $(C_{L^{2} (Ω)}), C_{H_{0}^{1} (Ω)}))$ -uniform (w.r.t $σ \in Σ$ ) attractors for ${U_{σ} (t, τ)}_{σ \in Σ}$ .

At first, we have following lemma, which will be helpful to prove the existence of the $(C_{L^{2} (Ω)}, C_{H_{0}^{1} (Ω)})$ -uniform (w.r.t $σ \in Σ$ ) attractors for ${U_{σ} (t, τ)}_{σ \in Σ}$ .

Lemma 3.11.

Let f satisfy ( 1.2 )–( 1.3 ), $g (\cdot)$ subject to assumptions (I)–(II), $k (\cdot) \in L_{b}^{2} (R, L^{2} (Ω))$ and $ϕ \in C_{L^{2} (Ω)}$ . Assume further $k (\cdot) \in L_{n}^{2} (R, L^{2} (Ω))$ , then for any $ε > 0$ and any bounded set $B \subset L^{2} (Ω)$ , there exist $M > 0$ and $η > 0$ such that $\begin{array}{l} (3.29) & \int_{t}^{t + η} \int_{Ω (| u | ⩾ M)} | u |^{p - 1} {| {(| u | - M)}_{+} |}^{p - 1} d x d s ⩽ ε, \forall t ⩾ t_{0}, u (τ) \in B . \end{array}$

Proof.

Multiplying equation (1.1) by $| {(u - M)}_{+} |^{p - 2} {(u - M)}_{+}$ and being similar to the proof process in Theorem 3.9, from (3.18) we have $\begin{array}{l} (3.30) & \frac{d}{d t} {‖ {(u - M)}_{+} ‖}_{p}^{p} + \frac{c_{2} p}{4} \int_{Ω} | u |^{p - 1} {| {(u - M)}_{+} |}^{p - 1} ⩽ \frac{2 p}{c_{2}} L_{g}^{2} ‖ u_{t} ‖_{C_{L^{2} (Ω)}}^{2} + \frac{2 p}{c_{2}} {‖ k (t) ‖}_{2}^{2} . \end{array}$

Integrating (3.30) in $[t, t + η]$ with $t ⩾ t_{0}$ and $η > 0$ , we obtain that $\begin{array}{l} \frac{c_{2} p}{4} \int_{t}^{t + η} \int_{Ω} | u |^{p - 1} {| {(u - M)}_{+} |}^{p - 1} d s \\ ⩽ {‖ {(u (t) - M)}_{+} ‖}_{p}^{p} + \frac{2 p}{c_{2}} L_{g}^{2} \int_{t}^{t + η} ‖ u_{s} ‖_{C_{L^{2} (Ω)}}^{2} d s + \frac{2 p}{c_{2}} \int_{t}^{t + η} {‖ k (s) ‖}_{2}^{2} d s \\ ⩽ \int_{Ω (| u (t) | ⩾ M)} {| u (t) |}^{p} d x + \frac{2 p}{c_{2}} L_{g}^{2} \int_{t}^{t + η} ‖ u_{s} ‖_{C_{L^{2} (Ω)}}^{2} d s + \frac{2 p}{c_{2}} \int_{t}^{t + η} {‖ k (s) ‖}_{2}^{2} d s, \end{array}$ where $t_{0}$ comes from Theorem 3.9. Then combining with Definition 2.8, Corollary 3.6 and (3.27) in the proof process of Theorem 3.9, we can deduce that $\begin{array}{l} (3.31) & \int_{t}^{t + η} \int_{Ω (u ⩾ M)} | u |^{p - 1} {| {(u - M)}_{+} |}^{p - 1} d s ⩽ ε, \forall t ⩾ t_{0}, η > 0 . \end{array}$

On the other hand, taking $| {(u + M)}_{-} |^{p - 2} {(u + M)}_{-}$ instead of $| {(u - M)}_{+} |^{p - 2} {(u - M)}_{+}$ in the preceding proof, we deduce similarly that $\begin{array}{l} (3.32) & \int_{t}^{t + η} \int_{Ω (u ⩽ - M)} | u |^{p - 1} {| {(u + M)}_{-} |}^{p - 1} d s ⩽ ε, \forall t ⩾ t_{0}, η > 0 . \end{array}$

Combing with (3.31) and (3.32), we complete the proof. □

In the following, we obtain the $(C_{L^{2} (Ω)}, C_{H_{0}^{1} (Ω)})$ -uniform (w.r.t $σ \in Σ$ ) attractors for ${U_{σ} (t, τ)}_{σ \in Σ}$ .

Theorem 3.12.

Let f satisfy ( 1.2 )–( 1.3 ), $g (\cdot)$ subject to assumptions (I)–(II), $k (\cdot) \in L_{b}^{2} (R, L^{2} (Ω))$ and $ϕ \in C_{L^{2} (Ω)}$ . Assume further $k (\cdot) \in L_{n}^{2} (R, L^{2} (Ω))$ , then ${U_{σ} (t, τ)}_{σ \in Σ}$ has a $(C_{L^{2} (Ω)}, C_{H_{0}^{1} (Ω)})$ -uniform (w.r.t $σ \in Σ$ ) attractor, which is compact in $C_{H_{0}^{1} (Ω)}$ and uniformly (w.r.t $σ \in Σ$ ) attracts every bounded subset of $C_{L^{2} (Ω)}$ with the $C_{H_{0}^{1} (Ω)}$ -norm.

Proof.

By Lemma 3.8, we know that the family of processes ${U_{σ} (t, τ)}_{σ \in Σ}$ has a $(C_{L^{2} (Ω)}, C_{H_{0}^{1} (Ω)})$ -bounded uniformly (w.r.t $σ \in Σ$ ) absorbing set.

In the following, we will verify the family of the processes ${U_{σ} (t, τ)}_{σ \in Σ}$ satisfies the uniformly (w.r.t $σ \in Σ$ ) Condition (C).

Since ${(- Δ)}^{- 1}$ is a continuous compact operator in $L^{2} (Ω)$ , by the classical spectral theorem, there exists a sequence ${λ_{j}}_{j = 1}^{\infty}$ satisfying $\begin{array}{l} 0 < λ_{1} ⩽ λ_{2} ⩽ \dots ⩽ λ_{j} ⩽ \dots, λ_{j} \to + \infty as j \to + \infty, \end{array}$ and a family of elements ${ω_{j}}_{j = 1}^{\infty}$ of $D (- Δ)$ which are orthonormal in $L^{2} (Ω)$ such that $\begin{array}{l} - Δ ω_{j} = λ_{j} ω_{j}, \forall j \in N . \end{array}$

Let $V_{m} = span {ω_{1}, ω_{2}, \dots, ω_{m}}$ in V and $P_{m} : V \to V_{m}$ is an orthogonal projector.

For any $u \in D (- Δ)$ , we write $\begin{matrix} u = P_{m} u + (I - P_{m}) u = u_{1} + u_{2} . \end{matrix}$

Taking the inner product in $L^{2} (Ω)$ between equation (1.1) and $- Δ u_{2}$ , we have $\begin{array}{l} (3.33) & \frac{1}{2} \frac{d}{d t} ‖ \nabla u_{2} ‖_{2}^{2} + ‖ Δ u_{2} ‖_{2}^{2} = (f (u), - Δ u_{2}) + (g (u_{t}), - Δ u_{2}) + (k (t), - Δ u_{2}) . \end{array}$

By (1.3), we can get that $\begin{array}{l} | f (u) | ⩽ c (1 + | u |^{p - 1}), \end{array}$ then $\begin{array}{l} (f (u), - Δ u_{2}) & ⩽ | (f (u), - Δ u_{2}) | \\ ⩽ {‖ f (u) ‖}_{2}^{2} + \frac{1}{4} ‖ Δ u_{2} ‖_{2}^{2} \\ ⩽ c^{2} \int_{Ω} {(1 + | u |^{p - 1})}^{2} d x + \frac{1}{4} ‖ Δ u_{2} ‖_{2}^{2} \\ ⩽ 2 c^{2} | Ω | + 2 c^{2} \int_{Ω} | u |^{2 p - 2} d x + \frac{1}{4} ‖ Δ u_{2} ‖_{2}^{2} \\ = 2 c^{2} | Ω | + 2 c^{2} \int_{Ω (| u | ⩽ M)} | u |^{2 p - 2} d x + 2 c^{2} \int_{Ω (| u | ⩾ M)} | u |^{2 p - 2} d x + \frac{1}{4} ‖ Δ u_{2} ‖_{2}^{2} \\ (3.34) & ⩽ 2 c^{2} | Ω | + 2 c^{2} | M |^{2 p - 2} | Ω | + 2 c^{2} \int_{Ω (| u | ⩾ M)} | u |^{2 p - 2} d x + \frac{1}{4} ‖ Δ u_{2} ‖_{2}^{2} . \end{array}$

Moreover, by the Hölder and Young inequalities, the assumption (II), we obtain that $\begin{array}{l} (g (u_{t}), - Δ u_{2}) + (k (t), - Δ u_{2}) & ⩽ {‖ g (u_{t}) ‖}_{2} ‖ Δ u_{2} ‖_{2} + {‖ k (t) ‖}_{2} ‖ Δ u_{2} ‖_{2} \\ ⩽ 2 {‖ g (u_{t}) ‖}_{2}^{2} + 2 {‖ k (t) ‖}_{2}^{2} + \frac{1}{4} ‖ Δ u_{2} ‖_{2}^{2} \\ (3.35) & ⩽ 2 L_{g}^{2} ‖ u_{t} ‖_{C_{L^{2} (Ω)}}^{2} + 2 {‖ k (t) ‖}_{2}^{2} + \frac{1}{4} ‖ Δ u_{2} ‖_{2}^{2} . \end{array}$

Combining with (3.33)–(3.35) and applying the Poincaré inequality, we can deduce that $\begin{array}{l} \frac{d}{d t} ‖ \nabla u_{2} ‖_{2}^{2} + λ_{m + 1} ‖ \nabla u_{2} ‖_{2}^{2} ⩽ & 4 c^{2} | Ω | (1 + | M |^{2 p - 2}) + 4 c^{2} \int_{Ω (| u | ⩾ M)} | u |^{2 p - 2} d x \\ (3.36) & + 4 L_{g}^{2} ‖ u_{t} ‖_{C_{L^{2} (Ω)}}^{2} + 4 {‖ k (t) ‖}_{2}^{2} \end{array}$

Applying the Gronwall lemma to (3.36) and integrating it in $[t_{0}, t]$ , we arrive at $\begin{array}{l} {‖ \nabla u_{2} (t) ‖}_{2}^{2} ⩽ & {‖ \nabla u_{2} (t_{0}) ‖}_{2}^{2} e^{- λ_{m + 1} (t - t_{0})} + 4 c^{2} | Ω | (1 + | M |^{2 p - 2}) \int_{t_{0}}^{t} e^{- λ_{m + 1} (t - s)} d s \\ + 4 c^{2} \int_{t_{0}}^{t} e^{- λ_{m + 1} (t - s)} \int_{Ω (| u | ⩾ M)} | u |^{2 p - 2} d x d s \\ (3.37) & + 4 L_{g}^{2} \int_{t_{0}}^{t} e^{- λ_{m + 1} (t - s)} ‖ u_{s} ‖_{C_{L^{2} (Ω)}}^{2} d s + 4 \int_{t_{0}}^{t} e^{- λ_{m + 1} (t - s)} {‖ k (s) ‖}_{2}^{2} d s \end{array}$

Now, we estimate each term on the right of inequality (3.37).

At first, from Lemma 3.8 and let $t_{1} = \frac{1}{λ_{m + 1}} ln (6 ρ_{1}^{2} / ε) + t_{0}$ , then $t ⩾ t_{1}$ implies that $\begin{array}{l} (3.38) & {‖ \nabla u_{2} (t_{0}) ‖}_{2}^{2} e^{- λ_{m + 1} (t - t_{0})} ⩽ {‖ \nabla u (t_{0}) ‖}_{2}^{2} e^{- λ_{m + 1} (t - t_{0})} ⩽ \frac{ε}{6} . \end{array}$

Then, for the $ε > 0$ given above, we can take $m + 1$ large enough such that $\begin{array}{l} (3.39) & 4 c^{2} | Ω | (1 + | M |^{2 p - 2}) \int_{t_{0}}^{t} e^{- λ_{m + 1} (t - s)} d s ⩽ 4 c^{2} | Ω | (1 + | M |^{2 p - 2}) \frac{1}{λ_{m + 1}} ⩽ \frac{ε}{6} . \end{array}$

For the third term on the right of (3.37), we have $\begin{array}{l} \int_{Ω (| u | ⩾ M)} | u |^{2 p - 2} d x \\ = \int_{Ω (u ⩾ M)} | u |^{2 p - 2} d x + \int_{Ω (u ⩽ - M)} | u |^{2 p - 2} d x \\ = \int_{Ω (u ⩾ M)} | u |^{p - 1} | u - M + M |^{p - 1} d x + \int_{Ω (u ⩽ - M)} | u |^{p - 1} | u + M - M |^{p - 1} d x \\ ⩽ 2^{p - 2} \int_{Ω (u ⩾ M)} | u |^{p - 1} ({| {(u - M)}_{+} |}^{p - 1} + M^{p - 1}) d x \\ + 2^{p - 2} \int_{Ω (u ⩽ - M)} | u |^{p - 1} ({| {(u + M)}_{-} |}^{p - 1} + M^{p - 1}) d x \\ ⩽ 2^{p - 2} (\int_{Ω (u ⩾ M)} | u |^{p - 1} {| {(u - M)}_{+} |}^{p - 1} d x + \int_{Ω (u ⩽ - M)} | u |^{p - 1} {| {(u + M)}_{-} |}^{p - 1} d x) \\ + 2^{p - 2} M^{p - 1} \int_{Ω (| u | ⩾ M)} | u |^{p - 1} d x \\ (3.40) & ⩽ 2^{p - 2} \int_{Ω (| u | ⩾ M)} | u |^{p - 1} {| {(| u | - M)}_{+} |}^{p - 1} d x + 2^{p - 2} M^{p - 2} \int_{Ω (| u | ⩾ M)} | u |^{p} d x, \end{array}$ and $\begin{array}{l} \int_{t_{0}}^{t} e^{- λ_{m + 1} (t - s)} \int_{Ω (| u | ⩾ M)} | u |^{p - 1} {| {(u - M)}_{+} |}^{p - 1} d x d s \\ = (\int_{t - η}^{t} + \int_{t - 2 η}^{t - η} + \int_{t - 3 η}^{t - 2 η} + \dots) e^{- λ_{m + 1} (t - s)} \int_{Ω (| u | ⩾ M)} | u |^{p - 1} {| {(u - M)}_{+} |}^{p - 1} d x d s \\ ⩽ \int_{t - η}^{t} \int_{Ω (| u | ⩾ M)} | u |^{p - 1} {| {(u - M)}_{+} |}^{p - 1} d x d s \\ + e^{- λ_{m + 1} η} \int_{t - 2 η}^{t - η} \int_{Ω (| u | ⩾ M)} | u |^{p - 1} {| {(u - M)}_{+} |}^{p - 1} d x d s \\ + e^{- 2 λ_{m + 1} η} \int_{t - 3 η}^{t - 2 η} \int_{Ω (| u | ⩾ M)} | u |^{p - 1} {| {(u - M)}_{+} |}^{p - 1} d x d s + \dots \\ (3.41) & ⩽ \frac{1}{1 - e^{- λ_{m + 1} η}} \frac{ε}{12 c^{2} 2^{p}}, \end{array}$ where we have used (3.29) in Lemma 3.11. Moreover, thanks to (3.27) in Theorem 3.9 and let $m + 1$ large enough, we can get $\begin{array}{l} 4 c^{2} \int_{t_{0}}^{t} e^{- λ_{m + 1} (t - s)} (2^{p - 2} M^{p - 2} \int_{Ω (| u | ⩾ M)} | u |^{p} d x) d s \\ (3.42) & = c^{2} 2^{p} M^{p - 2} \int_{t_{0}}^{t} e^{- λ_{m + 1} (t - s)} (\int_{Ω (| u | ⩾ M)} | u |^{p} d x) d s ⩽ \frac{ε}{6} . \end{array}$

Finally, by (3.8) in Corollary 3.6 and Lemma 2.9, then let $m + 1$ large enough, we can see that $\begin{array}{l} (3.43) & 4 L_{g}^{2} \int_{t_{0}}^{t} e^{- λ_{m + 1} (t - s)} ‖ u_{s} ‖_{C_{L^{2} (Ω)}}^{2} d s ⩽ \frac{4}{λ_{m + 1}} L_{g}^{2} ρ_{0}^{2} ⩽ \frac{ε}{6}, \end{array}$ and $\begin{array}{l} (3.44) & 4 \int_{t_{0}}^{t} e^{- λ_{m + 1} (t - s)} {‖ k (s) ‖}_{2}^{2} d s ⩽ \frac{ε}{6} . \end{array}$

Combining with (3.37)–(3.44), we conclude that $\begin{array}{l} {‖ \nabla u_{2} (t) ‖}_{2}^{2} ⩽ ε, \forall t ⩾ t_{0}, u (τ) \in B . \end{array}$

In particular, putting $t + θ$ instead of t with $θ \in [- h, 0]$ , we obtain that $\begin{array}{l} max_{θ \in [- h, 0]} {‖ \nabla u_{2} (t + θ) ‖}_{2}^{2} ⩽ ε, \forall t - h ⩾ t_{0}, u (τ) \in B, \end{array}$ which together with Lemma 3.8 indicates that ${U_{σ} (t, τ)}_{σ \in Σ}$ satisfies the uniformly (w.r.t. $σ \in Σ$ ) Condition $(C)$ in $C_{H_{0}^{1} (Ω)}$ . By Lemma 2.13, we complete the proof. □

Footnotes

Acknowledgements

The authors would like to thank the referee for his/her helpful comments and suggestions.

This work was supported by the NSFC (Grants Nos. 11601522, 11801493), the Fundamental Research Funds for the Central Universities of China (Grants No. 17CX02048), the Province Natural Science Foundation of Hunan (Grants No. 2018JJ2416) and Hebei (Grants No. A2018203309).

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Uniform attractors for the non-autonomous reaction-diffusion equations with delays

Abstract

Keywords

1. Introduction

2. Preliminaries and abstract results

2.1. Preliminaries

Lemma 2.9 ([21]).

3.1. Existence and uniqueness results

Remark 3.4. In the following, in this paper, we always assume that { U σ ( t , τ ) } σ ∈ Σ is the family of processes defined by (3.1) and satisfies (3.2). 3.2. ( C L 2 ( Ω ) , C L 2 ( Ω ) ) -uniform attractors

Footnotes

Acknowledgements

References

Remark 3.4.
In the following, in this paper, we always assume that ${U_{σ} (t, τ)}_{σ \in Σ}$ is the family of processes defined by (3.1) and satisfies (3.2).

3.2. $(C_{L^{2} (Ω)}, C_{L^{2} (Ω)})$ -uniform attractors