Uniform attractors for multi-valued process generated by non-autonomous nonclassical diffusion equations with delay in unbounded domain without uniqueness of solutions

Abstract

In this article, the existence of a uniform attractor is proved for the multi-valued process generated by non-autonomous nonclassical diffusion equations (NDEs) with delays in unbounded domain without uniqueness of solutions, when the external force $g (x, t)$ belongs to the space $L_{C^{*}}^{2} (R, L^{2} (R^{N}))$ and is a translation bounded function, but is not a translation compact function.

Keywords

uniform attractors multi-valued process nonclassical diffusion equations with delay unbounded domain

1. Introduction

It is well known that in many cases the long-time behavior of dynamical systems generated by evolution equations of mathematical physics can be described naturally in terms of attractors of corresponding semigroups. The study of uniform attractor for non-autonomous infinite dimensional dynamical systems has attracted much attention and has made fast progress in recent decades, see, for instance, [3,7,8,10,15,19,20,22,26,29] and the references therein. Meanwhile, the asymptotical behavior of multi-valued non-autonomous infinite-dimensional dynamical systems has attracted much attention in mathematical literature, see, for example, [4,5,11,12,29–31] and the references therein.

In this paper, we investigate the asymptotic behavior of solutions to the following non-autonomous nonclassical diffusion equations on unbounded domain $R^{N}$ $\{\begin{matrix} \frac{\partial u}{\partial t} - Δ \frac{\partial u}{\partial t} - Δ u + λ u = f (x, u_{t}) + g (x, t), in R^{N} \times [τ, \infty), \\ u (x, t) = ϕ (x, t - τ), t \in [τ - h, τ], x \in R^{N}, \end{matrix}$ (1.1) where $λ > 0$ , ϕ is the initial data and the nonlinear term $f (x, u_{t} (x, t)) = F (x, u (x, t - ρ (t))) + \int_{- h}^{0} G (x, s, u (x, t + s)) d s .$ Now, we establish the following assumptions:

(i) there exist a positive constant $α_{2}$ , a positive scalar function $α_{1} \in L^{2} (R^{N})$ such that the functions $F \in C (R^{N} \times R; R)$ , $ρ \in C (R; [0, h])$ satisfy $\begin{array}{l} {| F (x, v) |}^{2} ⩽ {| α_{1} (x) |}^{2} + α_{2}^{2} | v |^{2}, \forall x \in R^{N}, v \in R; & (1.2) \\ | ρ^{'} (t) | ⩽ ρ_{*} < 1, \forall t \in R; & (1.3) \end{array}$

(ii) there exist positive scalar function $n \in L^{1} ([- h, 0])$ and $m \in L^{2} (R^{N})$ such that the function $G \in C (R^{N} \times [- h, 0] \times R; R)$ satisfies $| G (x, s, v) | ⩽ | m (x) | + n (s) | v |, \forall x \in R^{N}, s \in [- h, 0], v \in R,$ (1.4) where $\int_{- h}^{0} n (s) d s : = n > 0;$ (1.5)

(iii) the external force $g (x, t)$ belongs to the space $L_{C^{*}}^{2} (R, L^{2} (R^{N}))$ and is a translation bounded function, but is not a translation compact function.

The classical reaction–diffusion equation has strong background in mathematical physics, and it is very natural in many mathematical models. This problem arises in hydrodynamics and the heat transfer theory, such as heat transfer in a solid in contact with a moving fluid, thermoelastic distortion, diffusion phenomena, heat transfer in two media, problems in fluid dynamics etc. e.g. see [1,7,8,15,22,26,29]. In 1980, Aifantis in [1] pointed out that the classical reaction–diffusion equation does not contain each aspect of the reaction–diffusion problem, and it neglects viscidity, elasticity and pressure of medium in the process of solid diffusion and so on. In the sequel, Aifantis found out that the energy constitutional equation revealing the diffusion process is different along with the different property of the diffusion solid. Therefore, he constructed the mathematic model by some concrete examples, which contains viscidity, elasticity and pressure of medium, that is the nonclassical parabolic equation. The term $- Δ u_{t}$ denotes the pressure, viscoelasticity and memory, e.g. see [1,2,10,13,14]. For Eq. (1.1), without the variable delay term, the long-time behavior, especially the uniform attractor, pullback attractor and exponential attractors have been discussed by many authors in [3,10,16,17,21,23–25,28,32,33]. In [9], Hu and Wang proved the existence of pullback attractors for the non-autonomous nonclassical diffusion equation with delay in bounded domain. Caraballo in [6] proved the existence, uniqueness and asymptotic behavior of solutions for a nonclassical diffusion equation with delay in bounded domain. To our best knowledge, the existence of uniform attractors for multi-valued process generated by non-autonomous nonclassical diffusion equations with delay in unbounded domain has not been considered by predecessors.

Since Eq. (1.1) contains the term $- Δ u_{t}$ , this is essentially different from the classical reaction–diffusion equation. For example, the reaction–diffusion equation has some kind of “regularity”; e.g., although the initial datum only belongs to a weaker topology space, the solution will belong to a stronger topology space with higher regularity. However, for problem (1.1), because of $- Δ u_{t}$ , the solution has no higher regularity, i.e., if the initial datum only belongs to $C_{H^{1} (R^{N})}$ , then the solution is always in $C_{H^{1} (R^{N})}$ and has no higher regularity, which is similar to hyperbolic equations. This brings some difficulties in establishing the existence of uniform attractors for nonclassical diffusion equations. In addition, the delay term and unbounded domain also cause some difficulties to obtain the existence of uniform attractors, and some method used in a bounded domain cannot be directly employed in our case since the Sobolev embeddings are no longer compact.

To overcome these difficulties, we borrow the idea of the tail-estimated method introduced from Wang [27]; attractors for functional partial differential equations without uniqueness of solutions from Caraballo [4] and Wang [29]; and the new method to check the asymptotical upper-semicompactness of non-autonomous multi-valued delay dynamical systems from Wang [29,31], we prove the existence of uniform attractors in $C_{H^{1} (R^{N})}$ for multi-valued process generated by non-autonomous nonclassical diffusion equations with delay without uniqueness of solutions.

This paper is organized as following: In Section 1, we have expounded on research progress with regard to our research problem, and given some assumptions. In Section 2, we introduce some notations and functions spaces, and we give some useful lemmas. In Section 3, we prove the existence of uniform attractor in $C_{H^{1} (R^{N})}$ .

Our main result is Theorem 3.6.

2. Preliminaries

With the usual notation, hereafter let $| u |$ be the modular (or absolute value) of u, $| \cdot |$ be the norm of $L^{2} (R^{N}) = H$ , $∥ \cdot ∥$ be the norm of $H^{1} (R^{N}) = V$ . Let C the arbitrary positive constant, which may be different from line to line and even in the same line.

Next we iterate some definitions and abstract results concerning the uniform attractor, which is necessary to obtain our main results, please refer the reader to [7,8,15,22,26,29,31] for more details.

Let X be a complete metric space with metric $d_{X} (\cdot, \cdot)$ , and let $P (X)$ be the class of nonempty subsets of X. Denote by $H_{X}^{semi} (\cdot, \cdot)$ the Hausdorff semidistance between two nonempty subsets of a complete metric space X, which are defined by $H_{X}^{semi} (A, B) = sup_{a \in A} inf_{b \in B} d_{X} (a, b) .$ We denote by $N (A, r)$ the open neighborhood ${y \in X ∣ {dist}_{X} (y, A) < r}$ of radius $r > 0$ of a subset A of a complete metric space X.

Let $(Σ; d_{Σ})$ be a complete metric space. Σ will denote the symbol space underlying the dynamics.

Definition 2.1.
A family of mappings $U_{σ} (t, τ) : X \to P (X)$ , $t ⩾ τ$ , $τ \in R$ , is called to be a multi-valued process with $σ \in Σ$ if $\begin{matrix} U_{σ} (τ, τ) x = {x}, \forall τ \in R, x \in X; \\ U_{σ} (t, r) U_{σ} (r, τ) x = U_{σ} (t, τ) x for all τ ⩽ r ⩽ t, x \in X . \end{matrix}$ It is said to satisfy a translation identity with respect to a continuous semigroup ${T (h)}_{h \in R^{+}}$ of Σ onto itself if $U_{T (h) σ} (t, τ) = U_{σ} (t + h, τ + h), \forall h ⩾ 0; t ⩾ τ \in R .$ (2.1)
Definition 2.2.
A compact set $A \subset X$ is called the uniform attractor of the family of multi-valued processes ${U_{σ} (t, τ)}_{σ \in Σ}$ if
$A$ uniformly attracts every bounded subset B of X, i.e., for any fixed $τ \in R$ , $lim_{t \to + \infty} sup_{σ \in Σ} H_{X}^{semi} (U_{σ} (t + τ, τ) B, A) = 0,$

for another compact set $A^{'}$ satisfying (2.1), then $A \subset A^{'}$ .

The Kuratowski measure $k (A)$ of noncompactness of set A is defined by $k (A) = inf {δ > 0 ∣ A admits a finite cover by sets whose diameter ⩽ δ}$ .

The weak sequential closure ${\bar{A}}^{W S}$ of a subset A of a Banach space X is defined by ${\bar{A}}^{W S} = {x \in X ∣ \exists {x_{n}} \subset A, s.t. x_{n} ⇀ x, that is, x_{n} converges weakly to x} .$
Theorem 2.3 ([7,18,29]).

Let ${U_{σ} (t, τ)}_{σ \in Σ}$ be a family of multi-valued processes on a Banach space X satisfying the translation identity ( 2.1 ). Then the following statements are equivalent:

${U_{σ} (t, τ)}_{σ \in Σ}$ is uniformly dissipative, i.e., for any fixed $τ \in R$ , there exists a bounded subset $V$ of X so that for any bounded set $B \subset X$ , there exists a $T_{0} = T_{0} (B, τ) \in R^{+}$ , independent of $σ \in Σ$ , such that $U_{σ} (t + τ, τ) B \subset V, \forall σ \in Σ, t ⩾ T_{0},$ and uniformly ω-limit compact, i.e., for any fixed $τ \in R$ , and any bounded set B of X and any $ε > 0$ , there exists a $T_{1} = T_{1} (B, τ, ε) \in R^{+}$ , independent of $σ \in Σ$ , such that $k (⋃_{t ⩾ T_{1}} ⋃_{σ \in Σ} U_{σ} (t + τ, τ) B) ⩽ ε .$

${U_{σ} (t, τ)}_{σ \in Σ}$ has a unique uniform attractor $A = ω_{0, Σ} (V) = ω_{τ, Σ} (V)$ for $τ \in R$ , where $ω_{τ, Σ} (V) = ⋂_{T \in R^{+}} {\overline{⋃_{t ⩾ T} ⋃_{σ \in Σ} U_{σ} (t + τ, τ)}}^{W S} = ⋂_{T \in R^{+}} \overline{⋃_{t ⩾ T} ⋃_{σ \in Σ} U_{σ} (t + τ, τ)} .$

Let X be a Banach space with norm $∥ \cdot ∥_{X}$ , let $h > 0$ be a given positive number, which will denote the delay time, and we denote by $C_{X}$ a Banach space $C ((- h, 0]; X)$ with the norm $∥ ψ ∥_{C_{X}} : = sup_{θ \in [- h, 0]} {∥ ψ (θ) ∥}_{X} .$ Given $T > τ$ and $u : [τ - h, T) \to X$ , for each $t \in [τ, T)$ we denote by $u_{t}$ the function defined on $[- h, 0]$ by the relation $u_{t} (s) = u (t + s)$ , $s \in [- h, 0]$ .

The following theorem will be used to verify that the multi-valued process ${U (t, τ)}$ on $C_{X}$ is uniformly ω-limit compact.

Theorem 2.4 ([29]).

Let ${U_{σ} (t, τ)}_{σ \in Σ}$ be a multi-valued process on $C_{X}$ . Suppose that for each $τ \in R$ , every bounded subset B of $C_{X}$ and any $ε > 0$ , there exist $T = T_{B, τ, ε} \in R^{+}$ , independent of $σ \in Σ$ , a finite-dimensional subspace $X_{ε}$ of X and a $δ > 0$ such that

for each fixed $θ \in [- h, 0]$ , ${| ⋃_{t ⩾ T} ⋃_{σ \in Σ} ⋃_{u_{t} (\cdot) \in U_{σ} (t + τ, τ) B} P u (t + θ) |}_{X} is bounded,$

for all $t ⩾ T$ , $σ \in Σ$ , $u_{t} (\cdot) \in U_{σ} (t + τ, τ) B$ , $θ_{1}, θ_{2} \in [- h, 0]$ with $| θ_{2} - θ_{1} | < δ$ , ${| P (u (t + θ_{1}) - u (t + θ_{2})) |}_{X} < ε,$

for all $t ⩾ T$ , $σ \in Σ$ , $u_{t} (\cdot) \in U_{σ} (t + τ, τ) B$ , $sup_{θ \in [- h, 0]} {| (I - P) u (t + θ) |}_{X} < ε,$

where

P : X \to X_{ε}

is the canonical projector. Then

{U_{σ} (t, τ)}_{σ \in Σ}

is uniformly ω-limit compact in

C_{X}

Let $L_{b}^{2} (R; L^{2} (R^{N}))$ be a Banach space endowed with the norm $∥ g ∥_{L_{b}^{2} (R; L^{2} (R^{N}))} = {(sup_{t \in R} \int_{t}^{t + 1} {∥ g (s) ∥}^{2} d s)}^{\frac{1}{2}} .$

Definition 2.5.
Let X be a Banach space. A function $φ \in L_{b}^{2} (R, X)$ is said to satisfy condition $(C^{})$ if for any $ε > 0$ , there exists a final-dimensional subspace $X_{1}$ of X such that $sup_{t \in R} \int_{t}^{t + 1} {∥ (I - P_{m}) φ (x, s) ∥}^{2} d s ⩽ ε,$ where $P_{m} : X \to X_{1}$ is the canonical projector.

Let X be a Banach space, and denote by $L_{C^{}}^{2} (R, X)$ the set of all function satisfying condition $(C^{*})$ .
Theorem 2.6 ([18]).

If $φ \in L_{C^{*}}^{2} (R, X)$ , then for any $ε > 0$ and $τ \in R$ , we have $sup_{t ⩾ τ} \int_{τ}^{t} e^{- δ (t - s)} {∥ (I - P_{m}) φ (x, s) ∥}^{2} d s ⩽ ε,$ where $P_{m} : X \to X_{1}$ is the canonical projector, and $δ > 0$ .

Theorem 2.7 ([18]).

Let X be a Banach space, then $L_{C^{*}}^{2} (R, X)$ is a closed subspace of $L_{b}^{2} (R, X)$ .

Let $g_{0} \in L_{C^{*}}^{2} (R; L^{2} (R^{N}))$ , according to Theorem 2.7, $g_{0} \in L_{b}^{2} (R; L^{2} (R^{N}))$ , and define the hull $H (g_{0})$ of $g_{0}$ to be the closure of the ${g_{0} (τ + \cdot) : τ \in R}$ in $L_{b}^{2} (R; L^{2} (R^{N}))$ .

Theorem 2.8 ([7]).

For all $g \in H (g_{0})$ $∥ g ∥_{L_{b}^{2}}^{2} = sup_{t \in R} \int_{t}^{t + 1} {| g (s) |}^{2} d s ⩽ ∥ g_{0} ∥_{L_{b}^{2}}^{2} .$

Let us recall the following facts, which can be founded in Chepyzhov and Vishik [7].

Theorem 2.9.
Let X be a Banach space. A function $φ (s)$ is translation compact in $L_{loc}^{2} (R, X)$ if and only if
for any $r \in R$ , the set ${\int_{t}^{t + r} φ (s) d s ∣ t \in R}$ is precompact in X;

${lim}_{| l | \to 0} {sup}_{t \in R} \int_{t}^{t + 1} ∥ φ (s) - φ (s + l) ∥_{X}^{2} d s = 0$ .

Definition 2.10.
Let X be a Banach space. A function $φ (s)$ is translation bounded in $L_{loc}^{2} (R, X)$ , i.e., $∥ φ ∥_{L_{b}^{2} (R, X)}^{2} = sup_{t \in R} \int_{t}^{t + 1} {| φ (s) |}^{2} d s < \infty .$

3. Uniform attractors

Consider the following non-autonomous nonclassical diffusion equations on unbounded domain $R^{N}$ $\{\begin{matrix} \frac{\partial u}{\partial t} - Δ \frac{\partial u}{\partial t} - Δ u + λ u = f (x, u_{t}) + g (x, t), g \in H (g_{0}), \\ u (x, t) = ϕ (x, t - τ), t \in [τ - h, τ], x \in R^{N} . \end{matrix}$ (3.1)

We start with the following general existence of solutions which can be obtained by the standard Faedo–Galerkin methods (see [7,8,15,22,26]). Some techniques about the delays can be founded in [4] and the unbounded case can be founded in [6,17,27,29,33]. Here we only state the result:

3.1. Existence of solutions

Lemma 3.1.
Under the assumptions of ( 1.2 )–( 1.5 ), for each $τ \in R$ , and any $ϕ \in C_{V}$ , there is a solution $u (t)$ of ( 3.1 ) such that $u \in C ((τ - h, T]; V), \forall T > τ .$

According to Lemma 3.1, we can define a family of multi-valued process ${U_{g} (t, τ)}_{g \in H (g_{0})}$ on $C_{V}$ corresponding to (3.1) by $\begin{array}{rcl} U_{g} (t, τ) ϕ = {u_{t} (\cdot; τ, ϕ) ∣ u (\cdot) is a solution of (3.1) with ϕ \in C_{V} and symbol g \in H (g_{0})} . \end{array}$
3.2. Uniformly dissipative in $C_{V}$

Lemma 3.2.
Under the assumptions of ( 1.2 )–( 1.5 ), and $λ - α - n - \frac{3 α_{2}^{2} e^{α h}}{λ (1 - ρ^{})} > 0,$ (3.2) where λ is a large positive constant,* $n = \int_{- h}^{0} n (s) d s > 0$ , and $α \in (0, α^{})$ , $α^{} = min {α^{♯}, 2}$ and $α^{♯}$ is the solution of the equation $n e^{α h} = α$ , then for any $ϕ \in C_{V}$ , the solution $u_{t}$ of ( 3.1 ) satisfies $∥ u_{t} ∥_{C_{V}}^{2} ⩽ C e^{- α t} + C + C ∥ g_{0} ∥_{L_{b}^{2}}^{2} + C e^{(n e^{α h} - α) (t - τ)} .$ (3.3)
Proof.
Multiplying (3.1) by $u (t)$ , we infer $\begin{array}{l} \frac{d}{d t} (| \nabla u |^{2} + | u |^{2}) + 2 | \nabla u |^{2} + 2 λ | u |^{2} \\ = 2 F (t, u (t - ρ (t)), u) + 2 (\int_{- h}^{0} G (x, s, u (t + s)) d s, u) + 2 (g (x, t), u) . & (3.4) \end{array}$ Noting that (1.2), using Young’s inequality, for $μ, ν > 0$ , we infer $2 F (x, u (t - ρ (t)), u) ⩽ μ | u |^{2} + \frac{α_{2}^{2}}{μ} {| u (t - ρ (t)) |}^{2} + \frac{1}{μ} | α_{1} |^{2}$ (3.5) and $2 (g (x, t), u) ⩽ ν | u |^{2} + \frac{1}{ν} {| g (x, t) |}^{2} .$ (3.6) It follows from (3.4)–(3.6) that $\begin{array}{rclr} \frac{d}{d t} (| \nabla u |^{2} + | u |^{2}) & ⩽ & - 2 | \nabla u |^{2} + \frac{1}{μ} | α_{1} |^{2} + \frac{α_{2}^{2}}{μ} {| u (t - ρ (t)) |}^{2} + (μ + ν - 2 λ) | u |^{2} \\ + 2 (\int_{- h}^{0} G (x, s, u (t + s)) d s, u) + \frac{1}{μ} {| g (x, t) |}^{2} . & (3.7) \end{array}$ Multiplying (3.7) by $e^{α t}$ , we infer $\begin{array}{rclr} \frac{d}{d t} (e^{α t} (| \nabla u |^{2} + | u |^{2})) & = & α e^{α t} (| \nabla u |^{2} + | u |^{2}) + e^{α t} \frac{d}{d t} (| \nabla u |^{2} + | u |^{2}) \\ ⩽ & (α - 2) e^{α t} | \nabla u |^{2} + (μ + ν + α - 2 λ) e^{α t} | u |^{2} + \frac{1}{μ} | α_{1} |^{2} e^{α t} \\ + \frac{α_{2}^{2}}{μ} e^{α t} {| u (t - ρ (t)) |}^{2} + 2 e^{α t} (\int_{- h}^{0} G (x, s, u (t + s)) d s, u) \\ + \frac{1}{μ} e^{α t} {| g (x, t) |}^{2} . & (3.8) \end{array}$ Noting that $α < 2$ , now integrating (3.8) from τ to t, thus $\begin{array}{rclr} e^{α t} ({| \nabla u (t) |}^{2} + {| u (t) |}^{2}) & ⩽ & e^{α τ} ({| \nabla u (τ) |}^{2} + {| u (τ) |}^{2}) + \frac{1}{μ} | α_{1} |^{2} \int_{τ}^{t} e^{α s} d s \\ + \frac{α_{2}^{2}}{μ} \int_{τ}^{t} e^{α s} {| u (s - ρ (s)) |}^{2} d s + \frac{1}{μ} \int_{τ}^{t} e^{α s} {| g (x, s) |}^{2} d s \\ + 2 \int_{τ}^{t} e^{α s} (\int_{- h}^{0} G (x, r, u (t + r)) d r, u (s)) d s \\ + (μ + ν + α - 2 λ_{1}) \int_{τ}^{t} e^{α s} {| u (s) |}^{2} d s . & (3.9) \end{array}$ Using (1.3) and Young’s inequality, for $ε > 0$ , we get $\begin{array}{l} 2 | \int_{τ}^{t} e^{α s} (\int_{- h}^{0} G (x, r, u (t + r)) d r, u (s)) d s | \\ ⩽ 2 \int_{τ}^{t} e^{α s} (\int_{R^{N}} \int_{- h}^{0} (| m (x) | + n (r) | u (s + r) |) | u (s) | d r d x) d s \\ ⩽ 2 h \int_{τ}^{t} e^{α s} (\int_{R^{N}} | m (x) | | u (s) | d x) d s + 2 \int_{τ}^{t} e^{α s} (\int_{R^{N}} \int_{- h}^{0} n (r) | u (s + r) | | u (s) | d r d x) d s \\ ⩽ ε \int_{τ}^{t} e^{α s} {| u (s) |}^{2} d s + \frac{h^{2} | m |^{2}}{ε} \int_{τ}^{t} e^{α s} d s + n \int_{τ}^{t} e^{α s} ∥ u_{s} ∥_{C_{V}}^{2} d s + n \int_{τ}^{t} e^{α s} {| u (s) |}^{2} d s . & (3.10) \end{array}$ Noting that $ρ (s) \in [0, h]$ and the fact $\frac{1}{1 - ρ^{'} (s)} ⩽ \frac{1}{1 - ρ^{}}$ for all $s \in R$ . Setting $v = s - ρ (s)$ , we arrive at $\begin{array}{l} \frac{α_{2}^{2}}{μ} \int_{τ}^{t} e^{α s} {| u (s - ρ (s)) |}^{2} d s \\ ⩽ \frac{α_{2}^{2}}{μ} \int_{τ - h}^{t} \frac{e^{α h} e^{α v}}{1 - ρ^{}} {| u (v) |}^{2} d v \\ = \frac{α_{2}^{2} e^{α h}}{μ (1 - ρ^{})} (\int_{τ - h}^{τ} e^{α v} {| u (v) |}^{2} d v + \int_{τ}^{t} e^{α v} {| u (v) |}^{2} d v) \\ ⩽ \frac{α_{2}^{2} e^{α h + α τ} ∥ ϕ ∥_{C_{V}}^{2}}{μ α (1 - ρ^{})} + \frac{α_{2}^{2} e^{α h}}{μ (1 - ρ^{})} \int_{τ}^{t} e^{α v} {| u (v) |}^{2} d v . & (3.11) \end{array}$ It follows from (3.9)–(3.11) that $\begin{array}{l} e^{α t} ({| \nabla u (t) |}^{2} + {| u (t) |}^{2}) \\ ⩽ e^{α τ} ({| \nabla u (τ) |}^{2} + {| u (τ) |}^{2}) + (\frac{1}{μ} | α_{1} |^{2} + \frac{h^{2} | m |^{2}}{ε}) \int_{τ}^{t} e^{α s} d s \\ + n \int_{τ}^{t} e^{α s} ∥ u_{s} ∥_{C_{V}}^{2} d s + \frac{1}{μ} \int_{τ}^{t} e^{α s} {| g (x, s) |}^{2} d s + \frac{α_{2}^{2} e^{α h + α τ} ∥ ϕ ∥_{C_{V}}^{2}}{μ α (1 - ρ^{})} \\ + (μ + ν + α + ε + n + \frac{α_{2}^{2} e^{α h}}{μ (1 - ρ^{})} - 2 λ) \int_{τ}^{t} e^{α s} {| u (s) |}^{2} d s . & (3.12) \end{array}$ Choosing $μ = ν = ε = \frac{1}{3} λ$ , and noting (3.2), we have $μ + ν + α + ε + n + \frac{α_{2}^{2} e^{α h}}{μ (1 - ρ^{})} - 2 λ < 0 .$ Thus, $\begin{array}{l} e^{α t} ({| \nabla u (t) |}^{2} + {| u (t) |}^{2}) \\ ⩽ e^{α τ} ({| \nabla u (τ) |}^{2} + {| u (τ) |}^{2}) + (\frac{3}{λ} | α_{1} |^{2} + \frac{3 h^{2} | m |^{2}}{λ}) \int_{τ}^{t} e^{α s} d s \\ + n \int_{τ}^{t} e^{α s} ∥ u_{s} ∥_{C_{V}}^{2} d s + \frac{3}{λ} \int_{τ}^{t} e^{α s} {| g (x, s) |}^{2} d s + \frac{3 α_{2}^{2} e^{α h + α τ} ∥ ϕ ∥_{C_{V}}^{2}}{λ α (1 - ρ^{})} . & (3.13) \end{array}$ Setting $t + θ$ instead of t, where $θ \in [- h, 0]$ . Multiplying by $e^{- α t - α θ}$ , we infer $\begin{array}{l} {| \nabla u (t + θ) |}^{2} + {| u (t + θ) |}^{2} \\ ⩽ e^{α τ} e^{- α t - α θ} ({| \nabla u (τ) |}^{2} + {| u (τ) |}^{2}) + (\frac{3}{λ} | α_{1} |^{2} + \frac{3 h^{2} | m |^{2}}{λ}) e^{- α t - α θ} \int_{τ}^{t + θ} e^{α s} d s \\ + n e^{- α t - α θ} \int_{τ}^{t + θ} e^{α s} ∥ u_{s} ∥_{C_{V}}^{2} d s + \frac{3}{λ} e^{- α t - α θ} \int_{τ}^{t + θ} e^{α s} {| g (x, s) |}^{2} d s \\ + \frac{3 α_{2}^{2} e^{α h + α τ} e^{- α t - α θ} ∥ ϕ ∥_{C_{V}}^{2}}{λ α (1 - ρ^{})} \\ ⩽ e^{α τ} e^{- α t + α h} ({| \nabla u (τ) |}^{2} + {| u (τ) |}^{2}) + \frac{3}{λ α} | α_{1} |^{2} + \frac{3 h^{2} | m |^{2}}{λ α} \\ + n e^{- α t + α h} \int_{τ}^{t} e^{α s} ∥ u_{s} ∥_{C_{V}}^{2} d s + \frac{3}{λ} e^{- α t + α h} \int_{τ}^{t} e^{α s} {| g (x, s) |}^{2} d s + \frac{3 α_{2}^{2} e^{α τ} e^{- α t} ∥ ϕ ∥_{C_{V}}^{2}}{λ α (1 - ρ^{})} . & (3.14) \end{array}$ Hence, we obtain $\begin{array}{l} e^{α t} ∥ u_{t} ∥_{C_{V}}^{2} \\ ⩽ e^{α (τ + h)} ({| \nabla u (τ) |}^{2} + {| u (τ) |}^{2}) + (\frac{3}{λ α} | α_{1} |^{2} + \frac{3 h^{2} | m |^{2}}{λ α}) e^{α t} \\ + n e^{α h} \int_{τ}^{t} e^{α s} ∥ u_{s} ∥_{C_{V}}^{2} d s + \frac{3}{λ} e^{α h} \int_{τ}^{t} e^{α s} {| g (x, s) |}^{2} d s + \frac{3 α_{2}^{2} e^{α τ} ∥ ϕ ∥_{C_{V}}^{2}}{λ α (1 - ρ^{})} . & (3.15) \end{array}$ We denote $\begin{matrix} C_{1} = e^{α (τ + h)} ({| \nabla u (τ) |}^{2} + {| u (τ) |}^{2}), C_{2} = \frac{3}{λ α} | α_{1} |^{2} + \frac{3 h^{2} | m |^{2}}{λ α}, \\ C_{3} = n e^{α h}, C_{4} = \frac{3}{λ} e^{α h}, C_{5} = \frac{3 α_{2}^{2} e^{α τ} ∥ ϕ ∥_{C_{V}}^{2}}{λ α (1 - ρ^{})} . \end{matrix}$ Using the integral form of Gronwall Lemma, we infer $\begin{array}{rclr} ∥ u_{t} ∥_{C_{V}}^{2} & ⩽ & (C_{1} + C_{5}) e^{- α t} + C_{2} + C_{4} e^{- α t} \int_{τ}^{t} e^{α s} {| g (x, s) |}^{2} d s \\ + (C_{1} + C_{5}) e^{C_{3} t - α t - C_{3} τ} + \frac{C_{2} C_{3}}{α - C_{3}} \\ + C_{4} e^{C_{3} t - α t} \int_{τ}^{t} e^{- C_{3} s + α s} {| g (x, s) |}^{2} d s . & (3.16) \end{array}$ According to Theorem 2.8, we have $\begin{array}{rclr} C_{4} e^{- α t} \int_{τ}^{t} e^{α s} {| g (x, s) |}^{2} d s & ⩽ & C_{4} \sum_{i = 0}^{\infty} \int_{t - i - 1}^{t - i} e^{- α (t - s)} {| g (x, s) |}^{2} d s \\ ⩽ & C_{4} \sum_{i = 0}^{\infty} e^{- α i} \int_{t - i - 1}^{t - i} {| g (x, s) |}^{2} d s \\ ⩽ & \frac{C_{4}}{1 - e^{- α}} ∥ g_{0} ∥_{L_{b}^{2}}^{2} . & (3.17) \end{array}$ Similar to (3.17), we get $C_{4} e^{C_{3} t - α t} \int_{τ}^{t} e^{- C_{3} s + α s} {| g (x, s) |}^{2} d s ⩽ \frac{C_{4}}{1 - e^{α - C_{3}}} ∥ g_{0} ∥_{L_{b}^{2}}^{2} .$ (3.18) Noting that $C_{3} = n e^{α h} < α$ , it follows from (3.16)–(3.18) that $\begin{array}{rclr} ∥ u_{t} ∥_{C_{V}}^{2} & ⩽ & (C_{1} + C_{5}) e^{- α t} + C_{2} + \frac{C_{4}}{1 - e^{- α}} ∥ g_{0} ∥_{L_{b}^{2}}^{2} \\ + C_{6} e^{(C_{3} - α) (t - τ)} + \frac{C_{2} C_{3}}{α - C_{3}} + \frac{C_{4}}{1 - e^{α - C_{3}}} ∥ g_{0} ∥_{L_{b}^{2}}^{2}, & (3.19) \end{array}$ where $C_{6} = e^{α h} (| \nabla u (τ) |^{2} + | u (τ) |^{2}) + \frac{3 α_{2}^{2} ∥ ϕ ∥_{C_{V}}^{2}}{λ α (1 - ρ^{})}$ .

This completes the proof. □

According to Lemma 3.2, it is straightforward to see that the multi-valued process ${U_{σ} (t, τ)}_{g \in H (g_{0})}$ is uniformly dissipative in $C_{V}$ .
3.3. Estimates on the exterior of a ball

Lemma 3.3.
Under the assumptions of Lemma 3.2, for any fixed $τ \in R$ , any $ϵ > 0$ and every bounded set $B \subset C_{V}$ , there exist $K > 0$ and $T > 0$ such that $sup_{θ \in [- h, 0]} \int_{Ω_{K}^{C}} ({| u (t + θ) |}^{2} + {| \nabla u (t + θ) |}^{2}) d x ⩽ ϵ,$ (3.20) $\forall t ⩾ T$ , $g \in H (g_{0})$ , $ϕ \in B, u_{t} (\cdot) \in U_{g} (t, τ) ϕ$ , and $Ω_{K}^{C} = {x \in R^{N} ∣ | x | ⩾ K}$ .
Proof.
Choose a smooth function $θ (x)$ with $θ (x) = \{\begin{matrix} 0, & 0 ⩽ s ⩽ 1, \\ 1, & s ⩾ 2, \end{matrix}$ (3.21) where $0 ⩽ θ (s) ⩽ 1$ , $s \in R^{+}$ , and there is a constant c such that $| θ^{'} (s) | ⩽ c$ for $s \in R^{+}$ .

Multiplying (3.1) by $θ^{2} (\frac{| x |^{2}}{K^{2}}) u$ and integrating on $R^{N}$ , we obtain $\begin{array}{l} \frac{1}{2} \frac{d}{d t} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) (| \nabla u |^{2} + | u |^{2}) - \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) u Δ u + λ \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) | u |^{2} \\ = \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) F (x, u (t - ρ (t), x)) + \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) \int_{- h}^{0} G (x, s, u (t + s, x)) d x d s \\ + \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) u g . & (3.22) \end{array}$ Next, we bound each term in (3.22) one by one as follows $\int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) u Δ u = - \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) | \nabla u |^{2} - \int_{R^{N}} \frac{4 x}{K^{2}} θ (\frac{| x |^{2}}{K^{2}}) θ^{'} (\frac{| x |^{2}}{K^{2}}) u \nabla u$ (3.23) and by (3.21), we get $\begin{array}{rclr} | \int_{R^{N}} \frac{4 x}{K^{2}} θ (\frac{| x |^{2}}{K^{2}}) θ^{'} (\frac{| x |^{2}}{K^{2}}) u \nabla u | & ⩽ & \frac{c}{K} \int_{K ⩽ | x | ⩽ \sqrt{2} K} | u | | \nabla u | \\ ⩽ & \frac{c}{K} \int_{R^{N}} | u | | \nabla u | \\ ⩽ & \frac{c}{K} | u | | \nabla u | \\ ⩽ & \frac{C_{7}}{K}, & (3.24) \end{array}$ for all $t ⩾ τ + 1$ , where $C_{7}$ is dependent on $∥ ϕ ∥_{C_{V}}^{2}$ , $∥ g_{0} ∥_{L_{b}^{2}}^{2}$ , λ, α, and $ρ^{}$ .

Using Young’s inequality, for $μ > 0$ , we infer $| \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) u g | = | \int_{| x | ⩾ K} θ^{2} (\frac{| x |^{2}}{K^{2}}) u g | ⩽ \frac{μ}{2} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) | u |^{2} + \frac{1}{2 μ} \int_{| x | ⩾ k} | g |^{2} .$ (3.25) Noting that (1.2), using Young’s inequality, for $ν > 0$ , we infer $\begin{array}{l} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) F (x, u (t - ρ (t)), u) \\ ⩽ \frac{ν}{2} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) | u |^{2} + \frac{α_{2}^{2}}{2 ν} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) {| u (t - ρ (t)) |}^{2} + \frac{1}{2 ν} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) | α_{1} |^{2} . & (3.26) \end{array}$ Using (1.3) and Young’s inequality, for $ε > 0$ , we get $\begin{array}{l} | \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) \int_{- h}^{0} G (x, s, u (t + s)) u (t) d s d x | \\ ⩽ \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) \int_{- h}^{0} (| m (x) | + n (s) | u (t + s) |) | u (t) | d s d x \\ ⩽ \frac{ε}{2} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) | u |^{2} d x + \frac{h^{2}}{2 ε} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) | m |^{2} \\ + \frac{n}{2} sup_{s \in [- h, 0]} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) {| u (t + s) |}^{2} d x + \frac{n}{2} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) | u |^{2} d x . & (3.27) \end{array}$ It follows from (3.22)–(3.27) that $\begin{array}{l} \frac{1}{2} \frac{d}{d t} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) (| \nabla u |^{2} + | u |^{2}) + \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) | \nabla u |^{2} + λ \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) | u |^{2} \\ ⩽ \frac{C_{7}}{K} + (\frac{μ}{2} + \frac{ν}{2} + \frac{ε}{2} + \frac{n}{2}) \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) | u |^{2} + \frac{1}{2 μ} \int_{| x | ⩾ K} | g |^{2} \\ + \frac{α_{2}^{2}}{2 ν} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) {| u (t - ρ (t)) |}^{2} + \frac{1}{2 ν} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) | α_{1} |^{2} \\ + \frac{h^{2}}{2 ε} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) | m |^{2} + \frac{n}{2} sup_{s \in [- h, 0]} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) {| u (t + s) |}^{2} d x & (3.28) \end{array}$ for all $t ⩾ τ + 1$ . Thus $\begin{array}{l} \frac{d}{d t} (e^{α t} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) (| \nabla u |^{2} + | u |^{2})) \\ = α e^{α t} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) (| \nabla u |^{2} + | u |^{2}) + e^{α t} \frac{d}{d t} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) (| \nabla u |^{2} + | u |^{2}) \\ ⩽ (α - 2) e^{α t} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) | \nabla u |^{2} + (μ + ν + ε + n + α - 2 λ) e^{α t} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) | u |^{2} \\ + \frac{1}{μ} e^{α t} \int_{| x | ⩾ K} | g |^{2} + \frac{α_{2}^{2}}{ν} e^{α t} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) {| u (t - ρ (t)) |}^{2} + \frac{1}{ν} e^{α t} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) | α_{1} |^{2} \\ + \frac{h^{2}}{ε} e^{α t} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) | m |^{2} + n sup_{s \in [- h, 0]} e^{α t} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) {| u (t + s) |}^{2} d x + \frac{C_{7}}{K} e^{α t} . & (3.29) \end{array}$ Integrating from τ to t with $t ⩾ τ + 1$ , we infer $\begin{array}{l} e^{α t} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) ({| \nabla u (t) |}^{2} + {| u (t) |}^{2}) \\ ⩽ e^{α τ} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) ({| \nabla u (τ) |}^{2} + {| u (τ) |}^{2}) + \frac{1}{ν} \int_{τ}^{t} e^{α s} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) | α_{1} |^{2} d x d s \\ + (μ + ν + ε + n + α - 2 λ) \int_{τ}^{t} e^{α s} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) {| u (s) |}^{2} d x d s \\ + \frac{1}{μ} \int_{τ}^{t} e^{α s} \int_{| x | ⩾ K} | g |^{2} d x d s + \frac{α_{2}^{2}}{ν} \int_{τ}^{t} e^{α s} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) {| u (s - ρ (s)) |}^{2} d x d s \\ + \frac{h^{2}}{ε} \int_{τ}^{t} e^{α s} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) | m |^{2} d x d s + \frac{C_{7}}{K} \int_{τ}^{t} e^{α s} d x d s \\ + n \int_{τ}^{t} e^{α s} sup_{s \in [- h, 0]} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) {| u (t + s) |}^{2} d x d s . & (3.30) \end{array}$ Noting that $ρ (s) \in [0, h]$ and the fact $\frac{1}{1 - ρ^{'} (s)} ⩽ \frac{1}{1 - ρ^{}}$ for all $s \in R$ . Setting $v = s - ρ (s)$ , we infer $\begin{array}{l} \frac{α_{2}^{2}}{ν} \int_{τ}^{t} e^{α s} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) {| u (s - ρ (s)) |}^{2} d x d s \\ ⩽ \frac{α_{2}^{2}}{ν} \int_{τ - h}^{t} \frac{e^{α h} e^{α v}}{1 - ρ^{}} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) {| u (v) |}^{2} d x d v \\ ⩽ \frac{α_{2}^{2} e^{α h}}{ν (1 - ρ^{})} (\int_{τ - h}^{τ} e^{α v} \int_{| x | ⩾ K} {| u (v) |}^{2} d x d v + \int_{τ}^{t} e^{α v} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) {| u (v) |}^{2} d x d v) . & (3.31) \end{array}$ Choosing $μ = ν = ε = \frac{1}{3} λ$ , and noting (3.2), we arrive at $μ + ν + α + ε + n + \frac{α_{2}^{2} e^{α h}}{μ (1 - ρ^{})} - 2 λ < 0 .$ Thus, for $t ⩾ τ + 1$ , we get $\begin{matrix} e^{α t} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) ({| \nabla u (t) |}^{2} + {| u (t) |}^{2}) \\ ⩽ e^{α τ} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) ({| \nabla u (τ) |}^{2} + {| u (τ) |}^{2}) + \frac{3}{λ} \int_{τ}^{t} e^{α s} \int_{| x | ⩾ K} | α_{1} |^{2} d x d s \\ + \frac{3}{λ} \int_{τ}^{t} e^{α s} \int_{| x | ⩾ K} | g |^{2} d x d s + \frac{3 α_{2}^{2} e^{α h}}{λ (1 - ρ^{})} \int_{τ - h}^{τ} e^{α s} \int_{| x | ⩾ K} {| u (s) |}^{2} d x d s \\ + \frac{3 h^{2}}{λ} \int_{τ}^{t} e^{α s} \int_{| x | ⩾ K} | m |^{2} d x d s + \frac{C_{7}}{K} e^{α t} \\ + n \int_{τ}^{t} e^{α s} sup_{s \in [- h, 0]} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) {| u (t + s) |}^{2} d x d s . \end{matrix}$ (3.32) Replacing t by $t + r$ , where $r \in [- h, 0]$ , and multiplying by $e^{- α (t + r)}$ , for $t ⩾ τ + 1$ , we get that $\begin{array}{l} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) ({| \nabla u (t + r) |}^{2} + {| u (t + r) |}^{2}) \\ ⩽ e^{α τ + α h} e^{- α t} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) ({| \nabla u (τ) |}^{2} + {| u (τ) |}^{2}) \\ + \frac{3 e^{α h}}{λ} e^{- α t} \int_{τ}^{t} e^{α s} \int_{| x | ⩾ K} | α_{1} |^{2} d x d s \\ + \frac{3 e^{α h}}{λ} e^{- α t} \int_{τ}^{t} e^{α s} \int_{| x | ⩾ K} | g |^{2} d x d s \\ + \frac{3 α_{2}^{2} e^{α h}}{λ (1 - ρ^{})} e^{- α t} \int_{τ - h}^{τ} e^{α s} \int_{| x | ⩾ K} {| u (s) |}^{2} d x d s \\ + \frac{3 h^{2} e^{α h}}{λ} e^{- α t} \int_{τ}^{t + r} e^{α s} \int_{| x | ⩾ K} | m |^{2} d x d s + \frac{C_{7}}{K} \\ + n e^{α h} \int_{τ}^{t} e^{α s} e^{- α t} sup_{s \in [- h, 0]} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) {| u (t + s) |}^{2} d x d s . & (3.33) \end{array}$ Hence, for $t ⩾ τ + 1$ , we infer $\begin{array}{l} e^{α t} sup_{s \in [- h, 0]} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) ({| \nabla u (t + r) |}^{2} + {| u (t + r) |}^{2}) d x \\ ⩽ e^{α τ + α h} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) ({| \nabla u (τ) |}^{2} + {| u (τ) |}^{2}) d x \\ + \frac{3 e^{α h}}{λ} \int_{τ}^{t} e^{α s} \int_{| x | ⩾ K} | α_{1} |^{2} d x d s \\ + \frac{3 e^{α h}}{λ} e^{- α t} \int_{τ}^{t} e^{α s} \int_{| x | ⩾ K} | g |^{2} d x d s \\ + \frac{3 α_{2}^{2} e^{α h}}{λ (1 - ρ^{})} \int_{τ - h}^{τ} e^{α s} \int_{| x | ⩾ K} {| u (s) |}^{2} d x d s \\ + \frac{3 h^{2} e^{α h}}{λ} \int_{τ}^{t} e^{α s} \int_{| x | ⩾ K} | m |^{2} d x d s + \frac{C_{7}}{K} e^{α t} \\ + n e^{α h} \int_{τ}^{t} e^{α s} sup_{s \in [- h, 0]} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) {| u (t + s) |}^{2} d x d s . & (3.34) \end{array}$ Now, for any $ϵ > 0$ , we can choose K large enough such that $\begin{matrix} \frac{C_{7}}{K} e^{α t} ⩽ ϵ e^{α t}, & (3.35) \\ \frac{3 α_{2}^{2} e^{α h}}{λ (1 - ρ^{*})} \int_{τ - h}^{τ} e^{α s} \int_{| x | ⩾ K} {| u (s) |}^{2} d x d s ⩽ C e^{α τ} ∥ ϕ ∥_{C_{V}}^{2}, & (3.36) \\ \frac{3 e^{α h}}{λ} \int_{τ}^{t} e^{α s} \int_{| x | ⩾ K} | α_{1} |^{2} d x d s + \frac{3 h^{2} e^{α h}}{λ} \int_{τ}^{t} e^{α s} \int_{| x | ⩾ K} | m |^{2} d x d s ⩽ ϵ e^{α t}, & (3.37) \\ \frac{3 e^{α h}}{λ} e^{- α t} \int_{τ}^{t} e^{α s} \int_{| x | ⩾ K} | g |^{2} d x d s \\ ⩽ \frac{3 e^{α h}}{λ} \sum_{i = 0}^{\infty} e^{- α i} \int_{t - i - 1}^{t - i} \int_{| x | ⩾ K} {| g (x, s) |}^{2} d x d s \\ ⩽ \frac{3 e^{α h}}{λ (1 - e^{- α})} sup_{t \in R} \int_{t}^{t + 1} \int_{| x | ⩾ K} {| g (x, s) |}^{2} d x d s \\ ⩽ C ϵ . & (3.38) \end{matrix}$ It follows from (3.34)–(3.38) that $\begin{array}{l} e^{α t} sup_{s \in [- h, 0]} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) ({| \nabla u (t + r) |}^{2} + {| u (t + r) |}^{2}) d x \\ ⩽ e^{α τ + α h} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) ({| \nabla u (τ) |}^{2} + {| u (τ) |}^{2}) d x \\ + C ϵ + C e^{α τ} ∥ ϕ ∥_{C_{V}}^{2} + ϵ e^{α t} \\ + n e^{α h} \int_{τ}^{t} e^{α s} sup_{s \in [- h, 0]} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) {| u (t + s) |}^{2} d x d s . & (3.39) \end{array}$ Using the integral form Gronwall Lemma, we infer $\begin{array}{l} sup_{s \in [- h, 0]} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) ({| \nabla u (t + r) |}^{2} + {| u (t + r) |}^{2}) d x \\ ⩽ C e^{α τ} e^{- α t} ∥ ϕ ∥_{C_{V}}^{2} + ϵ + C e^{(C_{3} - α) t} e^{(α - C_{3}) τ} ∥ ϕ ∥_{C_{V}}^{2} + C ϵ . & (3.40) \end{array}$ Now, we can take t large enough such that $sup_{s \in [- h, 0]} \int_{R^{N}} θ^{2} (\frac{| x |^{2}}{K^{2}}) ({| \nabla u (t + r) |}^{2} + {| u (t + r) |}^{2}) d x ⩽ ϵ .$ (3.41)

This completes the proof. □
3.4. Uniform attractors in $C_{V}$

We denote $Ω_{K} = {x \in R^{N} ∣ | x | < K}$ , and let $χ_{Ω_{K}} = \{\begin{matrix} 1, & \forall x \in Ω_{K}, \\ 0, & \forall x \in Ω_{K}^{C}, \end{matrix}$ (3.42) and $χ_{Ω_{K}}^{C} = \{\begin{matrix} 1, & \forall x \in Ω_{K}^{C}, \\ 0, & \forall x \in Ω_{K} . \end{matrix}$ (3.43) Let $u (x, t) = v (x, t) + w (x, t)$ , where $v (x, t) = u (x, t) χ_{Ω_{K}}$ and $w (x, t) = u (x, t) χ_{Ω_{K}}^{C}$ . We decompose (3.1) as follows: $\{\begin{matrix} \frac{\partial v}{\partial t} - Δ \frac{\partial v}{\partial t} - Δ v + λ v \\ = F (x, v (t - ρ (t))) χ_{Ω_{K}} + \int_{- h}^{0} G (x, s, v (t + s)) χ_{Ω_{K}} d s \\ + g (x, t) χ_{Ω_{K}}, g \in H (g_{0}), \\ v (x, t) = ϕ (t - τ, x) χ_{Ω_{K}}, t \in [τ - h, τ], x \in R^{N}, \end{matrix}$ (3.44) and $\{\begin{matrix} \frac{\partial w}{\partial t} - Δ \frac{\partial w}{\partial t} - Δ w + λ w \\ = F (x, u (t - ρ (t))) - F (x, v (t - ρ (t))) χ_{Ω_{K}} \\ + \int_{- h}^{0} G (x, s, u (t + s)) d s - \int_{- h}^{0} G (x, s, v (t + s)) χ_{Ω_{K}} d s \\ + g (x, t) χ_{Ω_{K}}^{C}, g \in H (g_{0}), \\ w (x, t) = ϕ (t - τ, x) χ_{Ω_{K}}^{C}, t \in [τ - h, τ], x \in R^{N} . \end{matrix}$ (3.45)

According to Lemma 3.1, there exists a solution $v (t)$ to problem (3.44), and Eq. (3.45) has a solution $w (t) : = u (t) - v (t)$ .

Lemma 3.4.

Under the assumptions of Lemma 3.2, for any fixed $τ \in R$ , any $ϵ > 0$ and every bounded subset $B \subset C_{V}$ , there exist $T > 0$ and a finite-dimensional subspace $P (H_{0}^{1} (Ω_{K}))$ of $H_{0}^{1} (Ω_{K})$ and $δ > 0$ such that

for all $t ⩾ T + τ$ , $g \in H (g_{0})$ , $u_{t} (\cdot) \in U_{g} (t + τ, τ) B$ , $θ_{1}, θ_{2} \in [- h, 0]$ with $| θ_{2} - θ_{1} | < δ$ , ${| P (v (t + θ_{1}) - v (t + θ_{2})) |}_{H_{0}^{1} (Ω_{K})} < ϵ;$

for all $t ⩾ T + τ$ , $g \in H (g_{0})$ , $u_{t} (\cdot) \in U_{g} (t + τ, τ) B$ , $sup_{θ \in [- h, 0]} {| (I - P) v (t + θ) |}_{H_{0}^{1} (Ω_{K})} < ϵ,$

where

P : H_{0}^{1} (Ω_{K}) \to P (H_{0}^{1} (Ω_{K}))

is the canonical projector.

Proof.

We consider the operator $A = - Δ$ with Dirichlet boundary conditions. Since A is self-adjoint, positive operator and has a compact inverse, there exists a complete set of eigenvectors ${ω_{i}}_{i = 1}^{\infty}$ in $L^{2} (Ω_{K})$ , the corresponding eigenvalues ${λ_{i}}_{i = 1}^{\infty}$ satisfy $A ω_{i} = λ_{i} ω_{i}, 0 < λ_{1} ⩽ λ_{2} ⩽ \dots ⩽ λ_{i} ⩽ \dots \to + \infty, i \to + \infty .$ We set $V_{m} = span {ω_{1}, ω_{2}, \dots, ω_{m}}$ . $P_{m}$ is the orthogonal projection onto $V_{m}$ , and $Q_{m} = I - P_{m}$ is the orthogonal projection onto the orthogonal complement of $V_{m}$ , $v = P_{m} v + Q_{m} v = v^{1} + v^{2}$ . We decompose (3.44) as follows: $\{\begin{matrix} \frac{\partial v^{1}}{\partial t} - Δ \frac{\partial v^{1}}{\partial t} - Δ v^{1} + λ v^{1} \\ = P_{m} F (x, v^{1} (t - ρ (t))) χ_{Ω_{K}} + \int_{- h}^{0} P_{m} G (x, s, v^{1} (t + s, x)) χ_{Ω_{K}} d s \\ + P_{m} g (x, t) χ_{Ω_{K}}, g \in H (g_{0}), \\ v^{1} (x, t) = P_{m} ϕ (t - τ, x) χ_{Ω_{K}}, t \in [τ - h, τ], x \in R^{N}, \end{matrix}$ (3.46) and $\{\begin{matrix} \frac{\partial v^{2}}{\partial t} - Δ \frac{\partial v^{2}}{\partial t} - Δ v^{2} + λ v^{2} \\ = F (x, v (t - ρ (t))) χ_{Ω_{K}} - P_{m} F (x, v^{1} (t - ρ (t))) χ_{Ω_{K}} \\ + \int_{- h}^{0} G (x, s, v (t + s, x)) χ_{Ω_{K}} d s - \int_{- h}^{0} P_{m} G (x, s, v^{1} (t + s, x)) χ_{Ω_{K}} d s \\ + Q_{m} g (x, t) χ_{Ω_{K}}, g \in H (g_{0}), \\ v^{2} (x, t) = Q_{m} ϕ (t - τ, x) χ_{Ω_{K}}, t \in [τ - h, τ], x \in R^{N} . \end{matrix}$ (3.47) We divide the proof into two steps. Step 1.

We consider the finite-dimensional functional differential system (3.46).

Noting that $| Δ v^{1} |^{2} ⩽ λ_{m} | \nabla v^{1} |^{2} ⩽ λ_{m}^{2} | v^{1} |^{2}$ . Without loss of generality, we assume that $θ_{1}, θ_{2} \in [- h, 0]$ with $0 < θ_{1} - θ_{2} < 1$ . Then $\begin{array}{l} {| v^{1} (t + θ_{1}) - v^{1} (t + θ_{2}) |}_{H_{0}^{1} (Ω_{K})} \\ ⩽ \sqrt{λ_{m}} | v^{1} (t + θ_{1}) - v^{1} (t + θ_{2}) | \\ ⩽ \sqrt{λ_{m}} \int_{t + θ_{2}}^{t + θ_{1}} | \frac{d v_{1} (T)}{d T} | d T \\ ⩽ \sqrt{λ_{m}} \int_{t + θ_{2}}^{t + θ_{1}} (| Δ \frac{d v^{1} (T)}{d T} | + | Δ v^{1} (T) | + λ | v^{1} (T) | + P_{m} F (x, v^{1} (T - ρ (T))) χ_{Ω_{K}} \\ + \int_{- h}^{0} | G (x, s, v^{1} (T + s)) χ_{Ω_{K}} | d s + | P_{m} g (x, T) χ_{Ω_{K}} |) d T . & (3.48) \end{array}$ According to (3.3), for all $t ⩾ τ$ , we get $\begin{array}{l} \int_{t + θ_{2}}^{t + θ_{1}} (| Δ v^{1} (T) | + λ | v^{1} (T) |) d T \\ ⩽ (λ_{m} + λ) \int_{t + θ_{2}}^{t + θ_{1}} | v^{1} (T) | d T \\ ⩽ C \int_{t + θ_{2}}^{t + θ_{1}} {| v^{1} (T) |}^{2} d T + C (θ_{1} - θ_{2}) \\ ⩽ C (e^{- α θ_{2}} - e^{- α θ_{1}}) + C (e^{(C_{3} - α) θ_{2}} - e^{(C_{3} - α) θ_{1}}) + C (θ_{1} - θ_{2}) + C ∥ g_{0} ∥_{L_{b}^{2}}^{2} (θ_{1} - θ_{2}) . & (3.49) \end{array}$ Using (1.2), we infer $\begin{array}{l} \int_{t + θ_{2}}^{t + θ_{1}} F (x, v^{1} (T - ρ (T))) χ_{Ω_{K}} d T \\ ⩽ \int_{t + θ_{2}}^{t + θ_{1}} ({| F (x, v^{1} (T - ρ (T))) |}^{2} + C) d T \\ ⩽ \int_{t + θ_{2}}^{t + θ_{1}} (α_{2}^{2} {∥ u (T) ∥}_{C_{V}}^{2} + C) d T \\ ⩽ C (e^{- α θ_{2}} - e^{- α θ_{1}}) + C (e^{(C_{3} - α) θ_{2}} - e^{(C_{3} - α) θ_{1}}) + C (θ_{1} - θ_{2}) + C ∥ g_{0} ∥_{L_{b}^{2}}^{2} (θ_{1} - θ_{2}) . & (3.50) \end{array}$ According to (1.4), we obtain $\begin{array}{l} \int_{t + θ_{2}}^{t + θ_{1}} \int_{- h}^{0} | G (x, s, v^{1} (T + s)) χ_{Ω_{K}} | d s d T \\ ⩽ \int_{t + θ_{2}}^{t + θ_{1}} (\int_{- h}^{0} (| m (x) | + n (s) | v (s + T) |) d s) d T \\ ⩽ C (θ_{1} - θ_{2}) + \int_{t + θ_{2}}^{t + θ_{1}} \int_{- h}^{0} (C + C m (s) + C m (s) {| u (s + T) |}^{2}) d s d T \\ ⩽ C (e^{- α θ_{2}} - e^{- α θ_{1}}) + C (e^{(C_{3} - α) θ_{2}} - e^{(C_{3} - α) θ_{1}}) + C (θ_{1} - θ_{2}) + C ∥ g_{0} ∥_{L_{b}^{2}}^{2} (θ_{1} - θ_{2}) . & (3.51) \end{array}$ Noting that $g \in H (g_{0}) \subset L_{b}^{2} (R; L^{2} (R^{N}))$ , we have $\begin{array}{l} sup_{t \in R} \int_{t + θ_{2}}^{t + θ_{1}} | P_{m} g (x, T) χ_{Ω_{K}} | d T \\ ⩽ sup_{t \in R} \int_{t + θ_{2}}^{t + θ_{1}} ({(θ_{1} - θ_{2})}^{\frac{1}{2}} {| g (T) |}^{2} + \frac{1}{4 {(θ_{1} - θ_{2})}^{\frac{1}{2}}}) d T \\ ⩽ {(θ_{1} - θ_{2})}^{\frac{1}{2}} sup_{t \in R} \int_{t}^{t + θ_{1} - θ_{2}} {| g (T) |}^{2} d T + \frac{1}{4} {(θ_{1} - θ_{2})}^{\frac{1}{2}} \\ ⩽ {(θ_{1} - θ_{2})}^{\frac{1}{2}} sup_{t \in R} \int_{t}^{t + 1} {| g (T) |}^{2} d T + \frac{1}{4} {(θ_{1} - θ_{2})}^{\frac{1}{2}} \\ ⩽ {(θ_{1} - θ_{2})}^{\frac{1}{2}} ∥ g_{0} ∥_{L_{b}^{2}}^{2} + \frac{1}{4} {(θ_{1} - θ_{2})}^{\frac{1}{2}} . & (3.52) \end{array}$ Now, it only remains to estimate the bound of $\sqrt{λ_{m}} \int_{t + θ_{2}}^{t + θ_{1}} | Δ \frac{d v^{1} (T)}{d T} | d T$ . Taking the inner product of (3.46) with $v^{1}$ , $v_{T}^{1}$ , $- Δ v^{1}$ and $- Δ v_{T}^{1}$ , respectively, then we infer $\begin{array}{l} \frac{d}{d T} ({| \nabla v^{1} |}^{2} + {| v^{1} |}^{2}) + 2 {| \nabla v^{1} |}^{2} + 2 λ {| v^{1} |}^{2} \\ = 2 (P_{m} F (T, v (T - ρ (T))) χ_{Ω_{K}}, v^{1}) \\ + 2 (\int_{- h}^{0} P_{m} G (x, s, v^{1} (T + s)) χ_{Ω_{K}} d s, v^{1}) + 2 (P_{m} g (x, T) χ_{Ω_{K}}, v^{1}), & (3.53) \\ \frac{d}{d T} ({| \nabla v^{1} |}^{2} + λ {| v^{1} |}^{2}) + 2 {| \nabla v_{T}^{1} |}^{2} + 2 {| v_{T}^{1} |}^{2} \\ = 2 (P_{m} F (T, v (T - ρ (T))) χ_{Ω_{K}}, v_{T}^{1}) \\ + 2 (\int_{- h}^{0} P_{m} G (x, s, v^{1} (T + s)) χ_{Ω_{K}} d s, v_{T}^{1}) + 2 (P_{m} g (x, T) χ_{Ω_{K}}, v_{T}^{1}), & (3.54) \\ \frac{d}{d T} ({| \nabla v^{1} |}^{2} + {| Δ v^{1} |}^{2}) + 2 {| Δ v^{1} |}^{2} + 2 λ {| \nabla v^{1} |}^{2} \\ = 2 (P_{m} F (T, v (T - ρ (T))) χ_{Ω_{K}}, - Δ v^{1}) \\ + 2 (\int_{- h}^{0} P_{m} G (x, s, v^{1} (T + s)) χ_{Ω_{K}} d s, - Δ v^{1}) + 2 (P_{m} g (x, T) χ_{Ω_{K}}, - Δ v^{1}) & (3.55) \end{array}$ and $\begin{array}{l} \frac{d}{d T} ({| Δ v^{1} |}^{2} + λ {| \nabla v^{1} |}^{2}) + 2 {| \nabla v_{T}^{1} |}^{2} + 2 {| Δ v_{T}^{1} |}^{2} \\ = 2 (P_{m} F (T, v (T - ρ (T))) χ_{Ω_{K}}, - Δ v_{T}^{1}) \\ + 2 (\int_{- h}^{0} P_{m} G (x, s, v^{1} (T + s)) χ_{Ω_{K}} d s, - Δ v_{T}^{1}) + 2 (P_{m} g (x, T) χ_{Ω_{K}}, - Δ v_{T}^{1}) . & (3.56) \end{array}$ Now, $(3.54) + (3.56) - (3.55) - λ (3.53)$ , using Young’s inequality, we infer $\begin{array}{l} 2 {| \nabla v_{T}^{1} |}^{2} + {| Δ v_{T}^{1} |}^{2} + {| v_{T}^{1} |}^{2} - 2 λ {| \nabla v^{1} |}^{2} - λ^{2} {| v^{1} |}^{2} - {| Δ v^{1} |}^{2} \\ = (P_{m} F (T, v (T - ρ (T))) χ_{Ω_{K}}, Δ v^{1} - Δ v_{T}^{1} + v_{T}^{1} - λ v^{1}) \\ + (\int_{- h}^{0} P_{m} G (x, s, v^{1} (T + s)) χ_{Ω_{K}} d s, Δ v^{1} - Δ v_{T}^{1} + v_{T}^{1} - λ v^{1}) \\ + (P_{m} g (x, T) χ_{Ω_{K}}, Δ v^{1} - Δ v_{T}^{1} + v_{T}^{1} - λ v^{1}) \\ ⩽ 4 {| P_{m} F (T, v (T - ρ (T))) χ_{Ω_{K}} |}^{2} + \frac{3}{4} ({| Δ v^{1} |}^{2} + {| Δ v_{T}^{1} |}^{2} + {| v_{T}^{1} |}^{2} + λ | v^{1} |^{2}) \\ + 4 \int_{- h}^{0} | P_{m} G (x, s, v^{1} (T + s)) χ_{Ω_{K}} | d s + 4 ∥ P_{m} g (x, T) χ_{Ω_{K}} ∥ . & (3.57) \end{array}$ Noting that $Ω_{K}$ is a bounded domain, using Poincaré inequality, then $\begin{array}{rclr} {| Δ v_{T}^{1} |}^{2} & ⩽ & 4 {| P_{m} F (T, v (T - ρ (T))) χ_{Ω_{K}} |}^{2} + C {| Δ v^{1} |}^{2} \\ + 4 \int_{- h}^{0} | P_{m} G (x, s, v^{1} (T + s)) χ_{Ω_{K}} | d s + 4 ∥ P_{m} g (x, T) χ_{Ω_{K}} ∥ . & (3.58) \end{array}$ According to (3.49)–(3.52), we infer $\begin{array}{l} \sqrt{λ_{m}} \int_{t + θ_{2}}^{t + θ_{1}} | Δ \frac{d v^{1} (T)}{d T} | d T \\ ⩽ C (e^{- α θ_{2}} - e^{- α θ_{1}}) + C (e^{(C_{3} - α) θ_{2}} - e^{(C_{3} - α) θ_{1}}) + C (θ_{1} - θ_{2}) + C ∥ g_{0} ∥_{L_{b}^{2}}^{2} (θ_{1} - θ_{2}) . & (3.59) \end{array}$ From (3.48)–(3.52), (3.59), for all $t ⩾ τ$ , $g \in H (g_{0})$ , $u_{t} \in U_{g} (t + τ, τ) B$ , we get that ${| v^{1} (t + θ_{1}) - v^{1} (t + θ_{2}) |}_{H_{0}^{1} (Ω_{K})} \to 0 as θ_{2} \to θ_{1} .$ (3.60) Step 2.

We consider the functional differential system (3.47).

Multiplying (3.47) by $- Δ v^{2}$ , we infer $\begin{array}{l} \frac{d}{d t} ({| \nabla v^{2} |}^{2} + {| Δ v^{2} |}^{2}) + 2 {| Δ v^{2} |}^{2} + 2 λ {| \nabla v^{2} |}^{2} \\ = 2 (F (t, v (t - ρ (t))) χ_{Ω_{K}} - P_{m} F (t, v^{1} (t - ρ (t))) χ_{Ω_{K}}, - Δ v^{2}) \\ + 2 (\int_{- h}^{0} G (x, s, v (t + s)) χ_{Ω_{K}} d s - \int_{- h}^{0} P_{m} G (x, s, v^{1} (t + s)) χ_{Ω_{K}} d s, - Δ v^{2}) \\ + 2 (Q_{m} g (x, t) χ_{Ω_{K}}, - Δ v^{2}) . & (3.61) \end{array}$ Using (1.2), we get $\begin{array}{l} 2 (F (t, v (t - ρ (t))) χ_{Ω_{K}} - P_{m} F (t, v^{1} (t - ρ (t))) χ_{Ω_{K}}, - Δ v^{2}) \\ ⩽ \frac{1}{2} {| Δ v^{2} |}^{2} + 16 α_{2}^{2} ∥ u_{t} ∥_{C_{V}}^{2} + 16 | α_{1} |^{2} . & (3.62) \end{array}$ By (1.4), we infer $\begin{array}{l} 2 (\int_{- h}^{0} G (x, s, v (t + s)) χ_{Ω_{K}} d s - \int_{- h}^{0} P_{m} G (x, s, v^{1} (t + s)) χ_{Ω_{K}} d s, - Δ v^{2}) \\ ⩽ 4 \int_{Ω} \int_{- h}^{0} (| m (x) | + n (s) | v (s + t) |) | Δ v^{2} | d s d x \\ ⩽ \frac{1}{2} {| Δ v^{2} |}^{2} + 16 m^{2} ∥ u_{t} ∥_{C_{V}}^{2} + 16 h^{2} | m |^{2}, & (3.63) \end{array}$ and $2 (Q_{m} g (x, t) χ_{Ω_{K}}, - Δ v^{2}) ⩽ \frac{1}{4} {| Δ v^{2} |}^{2} + 4 {∥ Q_{m} g (x, t) ∥}^{2} .$ (3.64) For $δ > 0$ small, it follows from (3.61)–(3.64) that $\begin{array}{l} \frac{d}{d t} ({| Δ v^{2} |}^{2} + {| \nabla v^{2} |}^{2}) + δ ({| \nabla v^{2} |}^{2} + {| Δ v^{2} |}^{2}) \\ ⩽ 16 h^{2} | m |^{2} + 16 | α_{1} |^{2} + 4 {∥ Q_{m} g (x, t) ∥}^{2} + 16 (m^{2} + | α_{1} |^{2}) ∥ u_{t} ∥_{C_{V}}^{2} . & (3.65) \end{array}$ Applying the Gronwall Lemma in the interval $[τ, t + θ]$ , we infer $\begin{matrix} sup_{θ \in [- h, 0]} ({| \nabla v^{2} (t + θ) |}^{2} + {| Δ v^{2} (t + θ) |}^{2}) \\ ⩽ e^{- δ (t - h - τ)} ({| \nabla v^{2} (τ) |}^{2} + {| Δ v^{2} (τ) |}^{2}) \\ + sup_{θ \in [- h, 0]} \int_{τ}^{t + θ} e^{- δ (t + θ - s)} (16 h^{2} | m |^{2} + 16 | α_{1} |^{2} + 16 (m^{2} + | α_{1} |^{2}) ∥ u_{s} ∥_{C_{V}}^{2}) d s \\ + 4 sup_{θ \in [- h, 0]} \int_{τ}^{t + θ} e^{- δ (t + θ - s)} {| Q_{m} g (x, s) |}^{2} d s . \end{matrix}$ (3.66) Noting that $g \in L_{C_{*}}^{2} (R, L^{2} (R^{N}))$ , from Theorem 2.6, for $ϵ > 0$ and $τ \in R$ , we get $4 sup_{θ \in [- h, 0]} \int_{τ}^{t + θ} e^{- δ (t + θ - s)} {| Q_{m} g (x, s) |}^{2} d s ⩽ ϵ .$ (3.67) Noting that $Ω_{K}$ is a bounded domain, using Poincaré inequality, then using (3.3), we can choose t and m large enough such that $sup_{θ \in [- h, 0]} \frac{1}{λ_{m + 1}} \int_{τ}^{t + θ} e^{- δ (t + θ - s)} (16 h^{2} | m |^{2} + 16 | α_{1} |^{2} + 16 (m^{2} + | α_{1} |^{2}) ∥ u_{s} ∥_{C_{V}}^{2}) d s ⩽ ϵ,$ (3.68) and $\frac{1}{λ_{m + 1}} e^{- δ (t - h - τ)} ({| \nabla v^{2} (τ) |}^{2} + {| Δ v^{2} (τ) |}^{2}) ⩽ ϵ .$ (3.69) Combining with (3.68)–(3.70), for $ϵ > 0$ and $τ \in R$ , when we choose t and m large enough, we infer ${∥ v_{t}^{2} ∥}_{C_{H_{0}^{1} (Ω_{K})}} = sup_{θ \in [- h, 0]} {| \nabla v^{2} (t + θ) |}_{H_{0}^{1} (Ω_{K})}^{2} ⩽ 3 ϵ .$ (3.70)

This completes the proof. □

For Eq. (3.45), similar to the proof of Lemma 3.3, we get the following conclusion.

Lemma 3.5.

Under the assumptions of Lemma 3.2, for any fixed $τ \in R$ , any $ϵ > 0$ and every bounded set $B \subset C_{V}$ , there exist $K > 0$ and $T > 0$ such that $sup_{θ \in [- h, 0]} \int_{Ω_{K}^{C}} ({| w (t + θ) |}^{2} + {| \nabla w (t + θ) |}^{2}) d x ⩽ ϵ,$ (3.71) $\forall t ⩾ T$ , $g \in H (g_{0})$ , $ϕ \in B$ , $u_{t} (\cdot) \in U_{g} (t, τ) ϕ$ , and $Ω_{K}^{C} = {x \in R^{N} ∣ | x | ⩾ K}$ .

Now, we state our main result.

Theorem 3.6.

Under the assumptions of Lemma 3.2, then the family of multi-valued processes ${U_{g} (t, τ)}_{g \in H (g_{0})}$ on $C_{V}$ possesses a unique uniform attractor $A_{C_{V}}$ .

Footnotes

Acknowledgements

The authors express their sincere thanks to the anonymous reviewer for his/her careful reading of the paper, giving helpful linguistic and mathematical comments. They also thank the editors for their kind help.

L. Bai was supported by the 2014 research funding of higher education of Gansu province project (2014A–142).

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