Upper semicontinuity of pullback attractors for nonclassical diffusion equations with delay

Abstract

In this paper, we mainly study the upper semicontinuity of pullback $D$ -attractors for a nonclassical diffusion equation with delay term $b (t, u_{t})$ which contains some hereditary characteristics. Under a critical nonlinearity f, a time-dependent force $g (t, x)$ with exponential growth and a delayed force term $b (t, u_{t})$ , using the asymptotic a priori estimate method, we prove the upper semicontinuity of pullback $D$ -attractor ${A_{ε} (t)}_{t \in R}$ to equation (1.1) with $ε \in [0, 1]$ .

Keywords

Nonclassical diffusion equations upper semicontinuity pullback attractors delay

1. Introduction

In this paper, we consider the following nonclassical diffusion equation with delay: $\begin{array}{l} (1.1) & \{\begin{matrix} \partial_{t} u (t, x) - ε Δ \partial_{t} u (t, x) - Δ u (t, x) + f (u (t, x)) = b (t, u_{t}) (x) + g (t, x) \\ in (τ, \infty) \times Ω, \\ u = 0 on (τ, \infty) \times \partial Ω, \\ u (τ, x) = u^{0} (x) τ \in R, x \in Ω, \\ u (τ + θ, x) = ϕ (θ, x) θ \in (- h, 0), x \in Ω, \end{matrix} \end{array}$ where $Ω \subset R^{N} (N ⩾ 3)$ is a bounded domain with smooth boundary $\partial Ω$ . $τ \in R$ , $u^{0} \in H_{0}^{1} (Ω)$ is the initial condition in τ and $ϕ \in L^{2} ([- h, 0]; L^{2} (Ω))$ is also the initial condition in $[τ - h, τ]$ , $h (> 0)$ is the length of the delay effects. And for each $t ⩾ τ$ , we denote by $u_{t}$ the function defined in $[- h, 0]$ by $u_{t} (θ) = u (t + θ), θ \in [- h, 0]$ . The nonlinearity f and the external forces b and g satisfy some specified conditions later, and $ε \in [0, 1]$ is a small positive perturbed parameter.

We assume the following conditions to hold (cf. [11]):

$(H 1)$ The nonlinear term $f \in C^{1} (R, R)$ with $f (0) = 0$ , satisfies $\begin{array}{l} (1.2) & f (u) u ⩾ - μ u^{2} - C_{1}, \\ (1.3) & f^{'} (u) ⩾ - l, \\ (1.4) & | f (u) | ⩽ C_{2} (1 + | u |^{p}), \\ (1.5) & lim_{| u | \to \infty} inf \frac{u f (u) - k F (u)}{u^{2}} ⩾ 0, \\ (1.6) & lim_{| u | \to \infty} inf \frac{F (u)}{u^{2}} ⩾ 0, \end{array}$ where $0 < p < min {\frac{N + 2}{N - 2}, 2 + \frac{4}{N}}$ , and μ, l, k, $C_{1}$ , $C_{2}$ are positive constants, $λ_{1} > 0$ is the first eigenvalue of $- Δ$ in Ω with the homogeneous Dirichlet condition such that $λ_{1} > max {μ, l}$ .

Let us denote by $\begin{array}{l} (1.7) & F (u) : = \int_{0}^{u} f (s) d s . \end{array}$

We infer from equation (1.5) and (1.6) that for any $δ > 0$ , there exist positive constants $C_{δ 1}, C_{δ 2}$ such that $\begin{array}{l} (1.8) & u f (u) - k F (u) + δ u^{2} + C_{δ 1} ⩾ 0 \forall u \in R, \\ (1.9) & F (u) + δ u^{2} + C_{δ 2} ⩾ 0 \forall u \in R . \end{array}$

$(H 2)$ With regard to the time-dependent external force term without delay, we assume that $g (t, x)$ , $\partial_{t} g (t, x) \in L_{l o c}^{2} (R; L^{2} (Ω))$ , and $g (t, x)$ satisfies $\begin{array}{l} (1.10) & \int_{- \infty}^{t} e^{σ s} {‖ g (s, x) ‖}^{2} d s < \infty, \forall t \in R, \end{array}$ where the positive constant σ satisfies $0 < σ < \bar{δ} < min {2 k, \frac{2 λ_{1}}{2 λ_{1} + 1}}$ .

$(H 3)$ We assume that the operator $b (\cdot, \cdot)$ is well-defined as $b : R \times L^{2} ((- h, 0); L^{2} (Ω)) \to L^{2} (Ω)$ which is a time-dependent external force with delay, and it satisfies:

for all $ϕ \in L^{2} ((- h, 0); L^{2} (Ω))$ , the function $t \in R \mapsto b (t, ϕ) \in L^{2} (Ω)$ is measurable;

$b (t, 0) = 0$ for all $t \in R$ ;

there exists a constant $L_{b} > 0$ such that for all $t \in R$ and $ϕ_{1}, ϕ_{2} \in L^{2} ((- h, 0); L^{2} (Ω))$ , $\begin{array}{l} (1.11) & ‖ b (t, ϕ_{1}) - b (t, ϕ_{2}) ‖ ⩽ L_{b} ‖ ϕ_{1} - ϕ_{2} ‖_{L^{2} ((- h, 0); L^{2} (Ω))}; \end{array}$

there exists a constant $C_{b} > 0$ such that for all $t ⩾ τ$ , and all $u, v \in L^{2} ([τ - h, t]; L^{2} (Ω))$ , $\begin{array}{l} (1.12) & \int_{τ}^{t} {‖ b (s, u_{s}) - b (s, v_{s}) ‖}^{2} d s ⩽ C_{b} \int_{τ - h}^{t} {‖ u (s) - v (s) ‖}^{2} d s; \end{array}$

for all $t ⩾ τ$ , and all $u, v \in L^{2} ([τ - h, t]; L^{2} (Ω))$ , $b (t, u_{t})$ also satisfies $\begin{array}{l} (1.13) & \int_{τ}^{t} e^{σ s} {‖ b (s, u_{s}) - b (s, v_{s}) ‖}^{2} d s ⩽ C_{b} \int_{τ - h}^{t} e^{σ s} {‖ u (s) - v (s) ‖}^{2} d s; \end{array}$

for all $t ⩾ τ$ , and all $u \in L^{2} ([τ - h, t]; L^{2} (Ω))$ , it holds that $\begin{array}{l} (1.14) & \int_{τ}^{t} {‖ \partial_{t} b (s, u_{s}) ‖}^{2} d s < \infty, \end{array}$ which means $\partial_{t} b (t, ϕ) \in L_{l o c}^{2} (R; L^{2} (Ω))$ .

Remark 1.1.
From above assumptions, for $T > τ$ , the function $t \in R \mapsto b (t, ϕ) \in L^{2} (Ω)$ is measurable and belongs to $L^{\infty} ((τ, T); L^{2} (Ω))$ .
Nonclassical reaction diffusion equation was derived from mechanics and heat conduction theory. During the last several decades, equations of this kind have been intensively applied in biology, chemistry and other fields [1,3,7,19]. The long-time behavior and well-posedness of the solutions of the nonclassical diffusion equations have been considered by many researchers, see [2,4,16,17,24]. In addition, the existence and upper semicontinuity of attractors of analogous dynamic equations have been analyzed by many scholars under distinct hypotheses. For example, the upper semicontinuity and regularity of global attractors for nonclassical diffusion equations were obatined in [22]. The upper semicontinuity of pullback attractors for a nonautonomous damped wave equation was studied in [18]. The existence of pullback attractors for 2D-Navier–Stokes model with delays was proved in [5].

As an effective tool for describing the asymptotic behavior of non-autonomous dynamical systems, pullback attractors are time-dependent families of compact sets which show the pullback sense when the initial time τ is taken to $- \infty$ . Actually, the large time behavior when the time t goes to $+ \infty$ driven by global attractors for autonomous dynamical system was also investigated by many researchers. Delay term is a kind of operators that expresses some kind of delay, memory or hereditary characteristics. For instance, we shall take into account not only the present state of the system, but also the history of the solutions when we want to control a system by applying external forces.

As far as we know, for the pullback attractors of nonclassical diffusion equations without delay, the limiting behavior of solutions to noncalssical diffusion equations also has been widely studied in recent years. In [2], the authors have proved the existence of pullback $D$ -attractor in the space $H_{0}^{1} (Ω)$ and the upper semicontinuity at $ε = 0$ . In [21], the euqation (1.1) without delay term $b (t, u_{t})$ in the phase space $H_{0}^{1} (Ω)$ was considered. It was proved that the upper semicontinuity of pullback attractors for a nonautonomous nonclassical diffusion equation with critical nonlinearity. The authors in [20] considered the upper simecontinuity of pullback attractors for a nonautonomous nonclassical diffusion equation whose external force has perturbation.

For the case with delay, asymptotic behavior of the equations similar to equation (1.1) has been deeply investigated during the last years. When $ε > 0$ , equation (1.1) is called the nonclassical diffusion equation with delay. In [11], the existence of pullback $D$ -attractors in $H_{0}^{1} (Ω) \times L^{2} ((- h, 0); L^{2} (Ω))$ with critical nonlinearity was considered. In [23], the existence of pullback $D$ -attractors in $C_{H_{0}^{1} (Ω))}$ with both critical nonlinearity and polynomial growth nonlinearity of arbitrary order was obtained. The long-time behavior of pullback attractors for the functional partial differential equation analogous to equaiton (1.1) without critial nonlinearity has been studied in [13]. When $ε = 0$ , equation (1.1) turns out to be the classical reaction-diffusion equation with delay. Similiar to nonclassical condition, the authors in [9] checked the existence of pullback $D$ -attractors in $C ([- h, 0]; L^{2} (Ω))$ with non-autonomous force in $H^{- 1}$ and delays under measurability conditions on driving delay term. In [12], the existence of pullback $D$ -attractors in $L^{2} (Ω) \times L^{2} ((- h, 0); L^{2} (Ω))$ was studied when the nonlinearity satisfies critical exponential growth condition.

Therefore, it is natural to examine the limiting behavior of solutions to equation (1.1) as perturbed parameter $ε \to 0$ when the exploration of upper semicontinuity of nonclassical diffusion equations with delay is in a blank state. The main difficulty is the handing of the time-delay term which forces us to consider the past history of the solution. However, for equation (1.1), because of the trem $- Δ \partial_{t} u$ , if the intial data belong to $H_{0}^{1} (Ω)$ , the solution is always in $H_{0}^{1} (Ω)$ and has no higher regularity, which is similar to the hyperbolic equation. In order to overcome the difficulities brought by delay term, we generalize the method for proving the upper semicontinuity of the pullback attractors proposed in [21] and use the priori estimation methods raised in [11] to verify the assumptions in Theorem 2.8, which implies the upper semicontinuity of pullback attractors. We also refer to [6,8,10].

The structure of the paper is the following. In Section 2, we shall recall some definitions and abstract results on pullback $D$ -attractors and upper semicontinuity (see [11] and [21]), respectively. In Section 3, by using the methods in [21], we shall prove the upper semicontinuity of equation (1.1) in the phase space $H_{0}^{1} (Ω) \times L^{2} ((- h, 0); L^{2} (Ω))$ .
2. Preliminaries

Throughout this section, we recall some basic concepts and results about pullback attractors and their upper semicontinuity, which will be used later.

For convenience, throughout this paper, let $| u |$ be the modular (or absolute value) of u. Let $‖ \cdot ‖$ and $⟨ \cdot, \cdot ⟩$ denote the norm and inner product in $L^{2} (Ω)$ , respectively; in the same way, let $‖ \nabla \cdot ‖$ and $⟨ \nabla \cdot, \nabla \cdot ⟩$ denote the norm and inner product in $H_{0}^{1} (Ω)$ , respectively. The norm in the Banach space $B$ will be denoted by $‖ \cdot ‖_{B}$ . C means any generic positive constant, which may be different from line to line and even in the same line. Let $X$ be a complete metric space with metric d. Denote by $P (X)$ the famliy of all nonempty subsets of $X$ , and suppose $D$ is a nonempty class of parameterized sets $\hat{D} = {D (t) : t \in R} \subset P (X)$ .

Definition 2.1 ([11,15]).

A two parameter family of mapping $U (t, τ) : X \to X$ , $t ⩾ τ$ , $τ \in R$ , is called to be a norm-to-weak continuous process if

(1) $U (τ, τ) x = x$ for all $τ \in R$ , $x \in X$ ;

(2) $U (t, s) U (s, τ) x = U (t, τ) x$ for all $t ⩾ s ⩾ τ$ , $τ \in R$ , $x \in X$ ;

(3) $U (t, τ) x_{n} ⇀ U (t, τ) x$ if $x_{n} \to x$ in $X$ .

Definition 2.2 ([11,14]).

A family of bounded set $\hat{B} = {B (t) : t \in R} \in D$ is called pullback $D$ -absorbing for the process ${U (t, τ)}$ if for any $t \in R$ and for any $\hat{D} \in D$ , there exists $τ_{0} (t, \hat{D}) ⩽ t$ such that $\begin{array}{l} U (t, τ) D (τ) \subset B (t) for all τ ⩽ τ_{0} (t, \hat{D}) . \end{array}$

Definition 2.3 ([14,20]).

A process ${U (t, τ)}$ is called pullback ω- $D$ -limit compact if for all $ε > 0$ and $\hat{D} \in D$ , there exists $τ_{0} (t, \hat{D}) ⩽ t$ such that $\begin{array}{l} K (⋃_{τ ⩽ τ_{0}} U (t, τ) D (τ)) ⩽ ε, \end{array}$ where $K$ is the Kuratowski measure of noncompactness of $B \in P (X)$ .

Definition 2.4 ([11,14]).

A family $\hat{A} = {A (t) : t \in R} \subset P (X)$ is said to be a pullback $D$ -attractor for ${U (t, τ)}$ if

(1) $A (t)$ is compact for all $t \in R$ ;

(2) $\hat{A}$ is invariant; i.e., $U (t, τ) A (τ) = A (t)$ , for all $t ⩾ τ$ ;

(3) $\hat{A}$ is pullback $D$ -attracting; i.e., $\begin{array}{l} lim_{τ \to - \infty} {dist}_{X} (U (t, τ) D (τ), A (t)) = 0, \end{array}$ for all $\hat{D} \in D$ and all $t \in R$ ;

(4) If ${C (t) : t \in R}$ is another family of closed attracting sets, then $A (t) \subset C (t)$ for all $t \in R$ .

Remark 2.1.
${dist}_{X} (A, B)$ denotes the Hausdorff semidistance between A and B which are bounded sets of $X$ , that is, ${dist}_{X} (A, B) = {sup}_{x \in A} {inf}_{y \in B} ‖ x - y ‖_{X}$ .

Theorem 2.5 ([11,14]).

Let ${U (t, τ)}$ be a norm-to-weak continuous process such that ${U (t, τ)}$ is pullback ω- $D$ -limit compact. If there exists a family of pullback $D$ -absorbing sets $\hat{B} = {B (t) : t \in R} \in D$ for the preocess ${U (t, τ)}$ , then there exists a pullback $D$ -attractor ${A (t) : t \in R}$ such that $\begin{array}{l} A (t) = ω (\hat{B}, t) = ⋂_{s ⩽ t} \overline{⋃_{τ ⩽ s} U (t, τ) B (τ)} . \end{array}$

Definition 2.6 ([21]).

Let $\hat{D} = {D (t)}_{t \in R}$ be a family of bounded subsets in $X$ . A process $U (\cdot, \cdot)$ is said to be pullback $D$ -asymptotically compact, if for any $t \in R$ , any sequence $τ_{n} \to - \infty$ and $x_{n} \in D (τ_{n})$ , the sequence ${U (t, τ_{n}) x_{n}}_{n \in N}$ is precompact in $X$ .

Lemma 2.7 ([20]).

$U (\cdot, \cdot)$ is pullback $D$ -asymptotic compact ⟺ $U (\cdot, \cdot)$ is pullback ω- $D$ -limit compact.

Theorem 2.8 ([21]).

Let $X$ be a Banach space with norm $‖ \cdot ‖_{X}$ . Assume for every $ε \in [0, μ] \subset R$ , process $U_{ε} (\cdot, \cdot)$ has a pullback $D_{ε}$ -absorbing family ${\hat{D}}_{ε} = {D_{ε} (t)}_{t \in R}$ and $U_{ε} (\cdot, \cdot)$ is pullback $D_{ε}$ -asymptotically compact in $X$ . Let $A_{ε} = {A_{ε} (t)}_{t \in R}$ be the pullback $D_{ε}$ -attractor in $X$ given by Theorem 2.5 . Suppose the following assumptions hold true:

(1) the pullback $D_{ε}$ -absorbing family ${\hat{D}}_{ε} = {D_{ε} (t)}_{t \in R}$ is independent of $ε \in [0, μ]$ , that is, there exists ${D (t)}_{t \in R}$ such that $\begin{array}{l} D_{ε} (t) = D (t), for all t \in R, and all ε \in [0, μ]; \end{array}$ (2) for any $τ < a$ and any bounded set $D \subset X$ , $\begin{array}{l} sup_{\begin{array}{c} t \in [a, b] \\ x \in D \end{array}} {‖ U_{ε} (t, τ) x - U_{ε_{0}} (t, τ) x ‖}_{X} \overset{ε \to ε_{0}}{⟶} 0, for any ε_{0} \in [0, μ]; \end{array}$ (3) for any $τ < a$ and any sequences ${t_{n}}_{n \in N} \subset [a, b]$ with $t_{n} \overset{n \to \infty}{⟶} t_{0}$ , ${x_{n}}_{n \in N} \subset X$ with $x_{n} \overset{n \to \infty}{⟶} x_{0}$ in $X$ , $\begin{array}{l} {‖ U_{ε} (t_{n}, τ) x_{n} - U_{ε} (t_{0}, τ) x_{0} ‖}_{X} \overset{n \to \infty}{⟶} 0, for any ε_{0} \in [0, μ] . \end{array}$ Then, $\begin{array}{l} (2.1) & lim_{ε \to ε_{0}} sup_{t \in [a, b]} {dist}_{X} (A_{ε} (t), A_{ε_{0}} (t)) = 0, for any ε_{0} \in [0, μ] \end{array}$ and $\begin{array}{l} (2.2) & ⋃_{t \in [a, b]} ⋃_{ε \in [0, μ]} A_{ε} (t) is precompact in X . \end{array}$

3. Upper semicontinuity

The existence and uniqueness of weak solution u to problem (1.1) (see, e.g., [11,12]) can be obtained by the usual Faedo–Galerkin approximation and a compactness method. Such solutions satisfy that: for any $ε \in [0, 1]$ and $[τ, T] \in R$ , $\begin{array}{l} u \in C ([τ, T]; H_{0}^{1} (Ω)), \frac{\partial u}{\partial t} \in L^{2} ((τ, T); H_{0}^{1} (Ω)), \end{array}$ with $u (t) = ϕ (t - τ)$ for $t \in [τ - h, τ]$ . To prove the main result of this section, we need the following lemmas.

Lemma 3.1 ([11]).

For any $τ \in R, T > τ$ , $u^{0} \in H_{0}^{1} (Ω)$ , $ϕ \in L^{2} ((- h, 0); L^{2} (Ω))$ and if there exist positive constants $η_{1}$ , $η_{2} < \frac{1}{2}$ such that $λ_{1} > max {μ, l} + \frac{η_{1} + η_{2}}{2} + \frac{C_{b}}{2 η_{1}}$ , then problem ( 1.1 ) has a unique weak solution u on $(τ, T)$ .

Invoking Lemma 3.1, we will apply the results in the phase space $Y : = H_{0}^{1} (Ω) \times L^{2} ((- h, 0); L^{2} (Ω))$ , which is a Hilbert space with the norm $\begin{array}{l} {‖ (u^{0}, ϕ) ‖}_{Y}^{2} = {‖ \nabla u^{0} ‖}^{2} + \int_{- h}^{0} {‖ ϕ (θ) ‖}^{2} d θ, \end{array}$ with a pair $(u^{0}, ϕ)$ of $Y$ .

Thus we can see that for any $ε \in [0, 1]$ , we can construct the process $U_{ε} (t, τ)$ with respect to problem (1.1) as follows: $\begin{array}{l} (u^{0}, ϕ) \mapsto U_{ε} (t, τ) (u^{0}, ϕ) = (u (t), u_{t}) for all t ⩾ τ and (u^{0}, ϕ) \in Y, \end{array}$ and the mapping $U_{ε} (t, τ) : Y \to Y$ is continuous.

Next, we need to consider the Hilbert space $\begin{array}{l} Y_{1} : = H_{0}^{1} (Ω) \times L^{2} ((- h, 0); H_{0}^{1} (Ω)), \end{array}$ with the norm $\begin{array}{l} {‖ (u^{0}, ϕ) ‖}_{Y_{1}}^{2} = {‖ \nabla u^{0} ‖}^{2} + \int_{- h}^{0} {‖ \nabla ϕ (θ) ‖}^{2} d θ . \end{array}$

We remark that when $t - τ ⩾ h$ , $U_{ε} (t, τ)$ maps $Y$ to $Y_{1}$ .

Lemma 3.2.

Let assumptions $(H 1)$ – $(H 3)$ be satisfied. Then for all t for which $t ⩾ τ + h$ and all $(u^{0}, ϕ) \in Y$ , we have following estimates: $\begin{array}{l} (3.1) & {‖ \nabla u (t) ‖}^{2} ⩽ C {e^{- σ (t - τ)} {‖ (u^{0}, ϕ) ‖}_{Y}^{\frac{2 N}{N - 2}} + 1 + e^{- σ t} \int_{- \infty}^{t} e^{σ s} {‖ g (s) ‖}^{2} d s}, \\ (3.2) & \begin{matrix} \int_{t - h}^{t} {‖ \nabla u (s) ‖}^{2} d s \\ ⩽ C β^{- 1} {e^{- σ (t - τ - h)} {‖ (u^{0}, ϕ) ‖}_{Y}^{\frac{2 N}{N - 2}} + e^{σ h} + e^{- σ (t - h)} \int_{- \infty}^{t} e^{σ s} {‖ g (s) ‖}^{2} d s}, \end{matrix} \\ (3.3) & \begin{matrix} e^{- σ t} \int_{τ}^{t} e^{σ s} ({‖ \partial_{t} u (s) ‖}^{2} + 2 ε {‖ \nabla \partial_{t} u (s) ‖}^{2}) d s \\ ⩽ C {e^{- σ (t - τ)} {‖ (u^{0}, ϕ) ‖}_{Y}^{\frac{2 N}{N - 2}} + 1 + e^{- σ t} \int_{- \infty}^{t} e^{σ s} {‖ g (s) ‖}^{2} d s}, \end{matrix} \end{array}$ for all $ε \in [0, 1]$ , where $β : = 2 \bar{δ} - 2 σ - \frac{6 C_{b}}{λ_{1}} > 0$ , $0 < σ < \bar{δ} < min {2 k, \frac{2 λ_{1}}{2 λ_{1} + 1}}$ , $C > 0$ is a positive constant which is independent of t, τ and ε.

Proof.

Multiplying (1.1) by $u + \partial_{t} u$ and integrating over Ω, we arrive at $\begin{array}{l} (3.4) & \begin{matrix} \frac{1}{2} \frac{d}{d t} ({‖ u (t) ‖}^{2} + (1 + ε) {‖ \nabla u (t) ‖}^{2}) + {‖ \nabla u (t) ‖}^{2} + {‖ \partial_{t} u (t) ‖}^{2} + ε {‖ \nabla \partial_{t} u (t) ‖}^{2} \\ + \int_{Ω} f (u) (u + \partial_{t} u) d x \\ = \int_{Ω} b (t, u_{t}) (u + \partial_{t} u) d x + \int_{Ω} g (t) (u + \partial_{t} u) d x . \end{matrix} \end{array}$ Using (1.8), the Cauchy–Schwarz and Young inequalities, we observe that $\begin{array}{l} \int_{Ω} b (t, u_{t}) (u + \partial_{t} u) d x + \int_{Ω} g (u + \partial_{t} u) d x \\ ⩽ \frac{3}{2} {‖ b (t, u_{t}) ‖}^{2} + \frac{3}{2} {‖ g (t) ‖}^{2} + {‖ u (t) ‖}^{2} + \frac{1}{2} {‖ \partial_{t} u (t) ‖}^{2}, \end{array}$ which gives $\begin{array}{l} (3.5) & \begin{matrix} \frac{1}{2} \frac{d}{d t} ({‖ u (t) ‖}^{2} + (1 + ε) {‖ \nabla u (t) ‖}^{2} + 2 \int_{Ω} F (u (t)) d x) + {‖ \nabla u (t) ‖}^{2} + \frac{1}{2} {‖ \partial_{t} u (t) ‖}^{2} \\ + ε {‖ \nabla \partial_{t} u (t) ‖}^{2} + k \int_{Ω} F (u (t)) d x \\ ⩽ \frac{3}{2} {‖ b (t, u_{t}) ‖}^{2} + \frac{3}{2} {‖ g (t) ‖}^{2} + (1 + δ) {‖ u (t) ‖}^{2} + C_{δ 1} | Ω | . \end{matrix} \end{array}$ By the Poincaré’s inequality, we get $\begin{array}{l} (3.6) & \begin{matrix} \frac{d}{d t} ({‖ u (t) ‖}^{2} + (1 + ε) {‖ \nabla u (t) ‖}^{2} + 2 \int_{Ω} F (u (t)) d x) + (2 - \frac{2 (1 + δ)}{λ_{1}}) {‖ \nabla u (t) ‖}^{2} \\ + {‖ \partial_{t} u (t) ‖}^{2} + 2 ε {‖ \nabla \partial_{t} u (t) ‖}^{2} + 2 k \int_{Ω} F (u (t)) d x \\ ⩽ 3 {‖ b (t, u_{t}) ‖}^{2} + 3 {‖ g (t) ‖}^{2} + 2 C_{δ 1} | Ω | . \end{matrix} \end{array}$ We can take δ small enough such that $\begin{array}{l} (3.7) & \bar{δ} ({‖ u (t) ‖}^{2} + (1 + ε) {‖ \nabla u (t) ‖}^{2}) ⩽ (2 - \frac{2 (1 + δ)}{λ_{1}}) {‖ \nabla u (t) ‖}^{2}, \end{array}$ where $0 < \bar{δ} < min {k, \frac{2 λ_{1}}{2 λ_{1} + 1}}$ . Then it follows that $\begin{array}{l} (3.8) & \frac{d}{d t} E (t) + \bar{δ} E (t) + {‖ \partial_{t} u (t) ‖}^{2} + 2 ε {‖ \nabla \partial_{t} u (t) ‖}^{2} ⩽ 3 {‖ b (t, u_{t}) ‖}^{2} + 3 {‖ g (t) ‖}^{2} + 2 C_{δ 1} | Ω |, \end{array}$ where $\begin{array}{l} (3.9) & E (t) = {‖ u (t) ‖}^{2} + (1 + ε) {‖ \nabla u (t) ‖}^{2} + 2 \int_{Ω} F (u (t)) d x . \end{array}$ Multiplying (3.8) by $e^{σ t}$ such that $0 < σ < \bar{δ}$ , we obtain $\begin{matrix} e^{σ t} \frac{d}{d t} E (t) + \bar{δ} e^{σ t} E (t) + e^{σ t} ({‖ \partial_{t} u (t) ‖}^{2} + 2 ε {‖ \nabla \partial_{t} u (t) ‖}^{2}) \\ ⩽ 3 e^{σ t} ({‖ b (t, u_{t}) ‖}^{2} + {‖ g (t) ‖}^{2} + C_{δ 1} | Ω |), \end{matrix}$ whence $\begin{matrix} \frac{d}{d t} (e^{σ t} E (t)) + e^{σ t} ({‖ \partial_{t} u (t) ‖}^{2} + 2 ε {‖ \nabla \partial_{t} u (t) ‖}^{2}) \\ ⩽ (σ - \bar{δ}) e^{σ t} E (t) + 3 e^{σ t} ({‖ b (t, u_{t}) ‖}^{2} + {‖ g (t) ‖}^{2} + C_{δ 1} | Ω |) . \end{matrix}$ Integrating the above inequality from τ to t, we have $\begin{matrix} E (t) + e^{- σ t} \int_{τ}^{t} e^{σ s} ({‖ \partial_{t} u (s) ‖}^{2} + 2 ε {‖ \nabla \partial_{t} u (s) ‖}^{2}) d s \\ ⩽ e^{- σ (t - τ)} E (τ) + (σ - \bar{δ}) e^{- σ t} \int_{τ}^{t} e^{σ s} E (s) d s + 3 e^{- σ t} \int_{τ}^{t} e^{σ s} {‖ b (s, u_{s}) ‖}^{2} d s \\ + 3 e^{- σ t} \int_{τ}^{t} e^{σ s} {‖ g (s) ‖}^{2} d s + 3 C_{δ 1} | Ω | σ^{- 1} . \end{matrix}$ Therefore, using (II) and (1.13), we obtain $\begin{array}{l} (3.10) & \begin{matrix} E (t) + e^{- σ t} \int_{τ}^{t} e^{σ s} ({‖ \partial_{t} u (s) ‖}^{2} + 2 ε {‖ \nabla \partial_{t} u (s) ‖}^{2}) d s \\ ⩽ e^{- σ (t - τ)} E (τ) + (σ - \bar{δ} e^{- σ t}) \int_{τ}^{t} e^{σ s} E (s) d s + 3 C_{b} e^{- σ (t - τ)} \int_{τ - h}^{τ} {‖ u (s) ‖}^{2} d s \\ + 3 C_{b} e^{- σ t} \int_{τ}^{t} e^{σ s} {‖ u (s) ‖}^{2} d s + 3 e^{- σ t} \int_{τ}^{t} e^{σ s} {‖ g (s) ‖}^{2} d s + 3 C_{δ 1} | Ω | σ^{- 1} . \end{matrix} \end{array}$ We use (1.6) in (3.9) to obtain by choosing δ small enough $\begin{array}{l} (3.11) & \begin{matrix} E (t) & ⩾ {‖ u (t) ‖}^{2} + (1 + ε) {‖ \nabla u (t) ‖}^{2} - 2 δ {‖ u (t) ‖}^{2} - 2 C_{δ 2} | Ω | \\ ⩾ \frac{1}{2} {‖ \nabla u (t) ‖}^{2} - 2 C_{δ 2} | Ω | . \end{matrix} \end{array}$ On the other hand, $\begin{array}{l} (3.12) & E (τ) = {‖ u (τ) ‖}^{2} + (1 + ε) {‖ \nabla u (τ) ‖}^{2} + 2 \int_{Ω} F (u (τ)) d x . \end{array}$ Thanks to (1.4), we get $\begin{array}{l} \int_{Ω} F (u) d x ⩽ C_{2} \int_{Ω} | u | d x + \frac{C_{2}}{p + 1} \int_{Ω} | u |^{p + 1} d x . \end{array}$ Applying the Hölder’s inequality, the Young inequality and the fact that $p + 1 ⩽ \frac{2 N}{N - 2}$ , we have $\begin{matrix} \int_{Ω} F (u) d x & ⩽ C \sqrt{| Ω |} {(\int_{Ω} | u |^{2} d x)}^{\frac{1}{2}} + \frac{C}{p + 1} \int_{Ω} | u |^{\frac{2 N}{N - 2}} d x + C \\ ⩽ C \sqrt{| Ω |} ‖ u ‖ + \frac{C}{p + 1} ‖ u ‖_{L^{\frac{2 N}{N - 2}} (Ω)}^{\frac{2 N}{N - 2}} + C . \end{matrix}$ By the Poincaré’s inequality, the Young inequality, $H_{0}^{1} (Ω) ↪ L^{\frac{2 N}{N - 2}} (Ω)$ and the fact that $1 < \frac{2 N}{N - 2}$ , we achieve $\begin{array}{l} (3.13) & \begin{matrix} \int_{Ω} F (u) d x & ⩽ C λ_{1}^{- 1} \sqrt{| Ω |} ‖ \nabla u ‖^{\frac{2 N}{N - 2}} + \frac{C}{p + 1} ‖ \nabla u ‖^{\frac{2 N}{N - 2}} + C \\ ⩽ C ‖ \nabla u ‖^{\frac{2 N}{N - 2}} + C, \end{matrix} \end{array}$ where C is a positive constant. Using (3.13) in (3.12), the Poincaré’s inequality, the Young inequality and the fact $2 < \frac{2 N}{N - 2}$ , we find that $\begin{array}{l} (3.14) & E (τ) ⩽ (λ_{1}^{- 1} + 1 + ε) {‖ \nabla u (τ) ‖}^{2} + C {‖ \nabla u (τ) ‖}^{\frac{2 N}{N - 2}} + C ⩽ C {‖ \nabla u (τ) ‖}^{\frac{2 N}{N - 2}} + C . \end{array}$ We substitute (3.11) and (3.14) in (3.10) to yield $\begin{matrix} \frac{1}{2} {‖ \nabla u (t) ‖}^{2} + e^{- σ t} \int_{τ}^{t} e^{σ s} ({‖ \partial_{t} u (s) ‖}^{2} + 2 ε {‖ \nabla \partial_{t} u (s) ‖}^{2}) d s \\ ⩽ C e^{- σ (t - τ)} {‖ \nabla u (τ) ‖}^{\frac{2 N}{N - 2}} + C e^{- σ (t - τ)} + 3 C_{b} e^{- σ (t - τ)} \int_{τ - h}^{τ} {‖ u (s) ‖}^{2} d s \\ + 3 C_{b} e^{- σ t} \int_{τ}^{t} e^{σ s} {‖ u (s) ‖}^{2} d s \\ + (σ - \bar{δ}) e^{- σ t} \int_{τ}^{t} e^{σ s} ({‖ u (s) ‖}^{2} + (1 + ε) {‖ \nabla u (s) ‖}^{2} + 2 \int_{Ω} F (u (s)) d x) d s \\ + 3 e^{- σ t} \int_{τ}^{t} e^{σ s} {‖ g (s) ‖}^{2} d s + 3 C_{δ 1} | Ω | σ^{- 1} + 2 C_{δ 2} | Ω | . \end{matrix}$ Then, for $2 \bar{δ} - 2 σ - \frac{6 C_{b}}{λ_{1}} > 0$ , noting the fact that $2 < \frac{2 N}{N - 2}$ and using (1.10) and the Young inequality, we deduce that $\begin{matrix} {‖ \nabla u (t) ‖}^{2} + (2 \bar{δ} - 2 σ - \frac{6 C_{b}}{λ_{1}}) e^{- σ t} \int_{τ}^{t} e^{σ s} {‖ \nabla u (s) ‖}^{2} d s + 2 e^{- σ t} \int_{τ}^{t} e^{σ s} ({‖ \partial_{t} u (s) ‖}^{2} \\ + 2 ε {‖ \nabla \partial_{t} u (s) ‖}^{2}) d s \\ ⩽ C e^{- σ (t - τ)} {‖ \nabla u (τ) ‖}^{\frac{2 N}{N - 2}} + C + C e^{- σ (t - τ)} ‖ ϕ ‖_{L^{2} ((- h, 0); L^{2} (Ω))}^{\frac{2 N}{N - 2}} + 6 e^{- σ t} \int_{τ}^{t} e^{σ s} {‖ g (s) ‖}^{2} d s \\ + 6 C_{δ 1} | Ω | σ^{- 1} + 4 C_{δ 2} | Ω | \\ ⩽ C {e^{- σ (t - τ)} ({‖ \nabla u^{0} ‖}^{\frac{2 N}{N - 2}} + ‖ ϕ ‖_{L^{2} ((- h, 0); L^{2} (Ω))}^{\frac{2 N}{N - 2}}) + 1 + e^{- σ t} \int_{τ}^{t} e^{σ s} {‖ g (s) ‖}^{2} d s} \\ ⩽ C {e^{- σ (t - τ)} {‖ (u^{0}, ϕ) ‖}_{Y}^{\frac{2 N}{N - 2}} + 1 + e^{- σ t} \int_{- \infty}^{t} e^{σ s} {‖ g (s) ‖}^{2} d s} . \end{matrix}$ Hence, for all $t ⩾ τ$ , we obtain (3.1), and $\begin{matrix} β e^{- σ t} \int_{τ}^{t} e^{σ s} {‖ \nabla u (s) ‖}^{2} d s \\ ⩽ C {e^{- σ (t - τ)} {‖ (u^{0}, ϕ) ‖}_{Y}^{\frac{2 N}{N - 2}} + 1 + e^{- σ t} \int_{- \infty}^{t} e^{σ s} {‖ g (s) ‖}^{2} d s}, \end{matrix}$ where $β : = 2 \bar{δ} - 2 σ - \frac{6 C_{b}}{λ_{1}} > 0$ . Furthermore, for $t - h ⩾ τ$ , we conclude $\begin{array}{l} \int_{τ}^{t} e^{σ s} {‖ \nabla u (s) ‖}^{2} d s ⩾ \int_{t - h}^{t} e^{σ s} {‖ \nabla u (s) ‖}^{2} d s ⩾ e^{σ (t - h)} \int_{t - h}^{t} {‖ \nabla u (s) ‖}^{2} d s, \end{array}$ as $[t - h, t] \subset [τ, t]$ . Therefore, for $t - h ⩾ τ$ , we obtatin (3.2). Besides, for all $t ⩾ τ$ , we derive $\begin{matrix} 2 e^{- σ t} \int_{τ}^{t} e^{σ s} ({‖ \partial_{t} u (s) ‖}^{2} + 2 ε {‖ \nabla \partial_{t} u (s) ‖}^{2}) d s \\ ⩽ C {e^{- σ (t - τ)} {‖ (u^{0}, ϕ) ‖}_{Y}^{\frac{2 N}{N - 2}} + 1 + e^{- σ t} \int_{- \infty}^{t} e^{σ s} {‖ g (s) ‖}^{2} d s}, \end{matrix}$ which is (3.3). □

Lemma 3.3.

Under assumptions $(H 1)$ – $(H 3)$ , for all $t \in (τ, T]$ and all $ε \in [0, 1]$ , the following estimate holds $\begin{array}{l} (3.15) & {‖ \partial_{t} u (t) ‖}^{2} + ε {‖ \nabla \partial_{t} u (t) ‖}^{2} ⩽ \frac{Q}{{(t - τ)}^{2}}, \end{array}$ provided that $‖ (u^{0}, ϕ) ‖_{Y}^{\frac{2 N}{N - 2}} ⩽ K$ , where the positive constant Q depends on τ, T and K, but is independent of ε.

Proof.

From Lemma 3.2 and (1.10), it follows that $\begin{array}{l} (3.16) & \int_{τ}^{T} ({‖ \partial_{t} u (s) ‖}^{2} + ε {‖ \nabla \partial_{t} u (s) ‖}^{2}) d s ⩽ M, \end{array}$ where $M > 0$ depends on τ, T and K. Differentiating equation (1.1) with respect to t and setting $v = \partial_{t} u$ , then v satisfies the following equality $\begin{array}{l} (3.17) & \partial_{t} v - ε Δ \partial_{t} v - Δ u + f^{'} (u) v = \partial_{t} g (t) + \partial_{t} b (t, u_{t}) . \end{array}$ Multiplying (3.17) by v and integrating it over Ω, using (1.3) and the Cauchy’s inequality, by the standard transformations, we obtain that $\begin{array}{l} (3.18) & \frac{d}{d t} I (t) ⩽ 2 l I (t) + λ_{1}^{- 1} {‖ \partial_{t} g (t, x) ‖}^{2} + λ_{1}^{- 1} {‖ \partial_{t} b (t, u_{t}) ‖}^{2}, \end{array}$ where $\begin{array}{l} (3.19) & I (t) = {‖ v (t) ‖}^{2} + ε {‖ \nabla v (t) ‖}^{2} . \end{array}$ If there exists a $ξ \in (τ, T]$ , such that the norm $‖ v (ξ) ‖^{2} + ε ‖ \nabla v (ξ) ‖^{2}$ is meaningful, we can apply the Gronwall inequality to (3.18) to find that for all $t \in [ξ, T]$ , $\begin{matrix} {‖ v (t) ‖}^{2} + ε {‖ \nabla v (t) ‖}^{2} \\ ⩽ Q ({‖ \partial_{t} u (ξ) ‖}^{2} + ε {‖ \nabla \partial_{t} u (ξ) ‖}^{2} + \int_{τ}^{t} {‖ \partial_{t} g (s, x) ‖}^{2} d s + \int_{τ}^{t} {‖ \partial_{t} b (s, u_{s}) ‖}^{2} d s), \end{matrix}$ where $Q = e^{2 l (t - ξ)} > 0$ depends on τ and T. Multiplying (3.18) by ${(t - τ)}^{2}$ , we obtain $\begin{array}{l} \frac{d}{d t} ({(t - τ)}^{2} I (t)) \\ ⩽ 2 l {(t - τ)}^{2} I (t) + 2 (t - τ) I (t) + λ_{1}^{- 1} {(t - τ)}^{2} {‖ \partial_{t} g (t, x) ‖}^{2} + λ_{1}^{- 1} {(t - τ)}^{2} {‖ \partial_{t} b (t, u_{t}) ‖}^{2} \\ (3.20) & ⩽ 2 (l + 1) {(t - τ)}^{2} I (t) + I (t) + λ_{1}^{- 1} {(t - τ)}^{2} {‖ \partial_{t} g (t, x) ‖}^{2} + λ_{1}^{- 1} {(t - τ)}^{2} {‖ \partial_{t} b (t, u_{t}) ‖}^{2} . \end{array}$ Applying the Gronwall inequality to (3.20), we get $\begin{array}{l} (3.21) & \begin{matrix} {(t - τ)}^{2} I (t) \\ ⩽ e^{\int_{ξ}^{t} (2 l + 1) d s} ({(ξ - τ)}^{2} I (ξ) + \int_{ξ}^{t} I (s) d s + λ_{1}^{- 1} \int_{ξ}^{t} {(s - τ)}^{2} {‖ \partial_{t} g (s, x) ‖}^{2} d s \\ + λ_{1}^{- 1} \int_{ξ}^{t} {(s - τ)}^{2} {‖ \partial_{t} b (s, u_{s}) ‖}^{2} d s) . \end{matrix} \end{array}$ Using (VI), (1.10) and (3.16), we conclude $\begin{array}{l} {(t - τ)}^{2} I (t) ⩽ Q, \end{array}$ where $Q > 0$ depends on τ, T and K, but is independent of ε. The proof is hence complete. □

Let $R$ be the set of functions $ρ : R \mapsto (0, + \infty)$ such that $\begin{array}{l} lim_{t \to - \infty} e^{σ t} ρ^{\frac{2 N}{N - 2}} (t) = 0 . \end{array}$ By $D$ we denote the class of all families $\hat{D} = {D (t) : t \in R} \subset P (Y)$ such that $D (t) \subset {\overline{B}}_{Y} (0, ρ (t))$ , for some $ρ \in R$ , where ${\overline{B}}_{Y} (0, ρ (t))$ denotes the closed ball in $Y$ centered at 0 with radius $ρ (t)$ .

Lemma 3.4.

Let assumptions $(H 1)$ – $(H 3)$ be satisfied. Then for every $ε \in [0, 1]$ , the process $U_{ε} (\cdot, \cdot)$ associated to equation ( 1.1 ) has a pullback $D$ -absorbing family $\hat{B} = {B (t)}_{t \in R}$ in $Y = H_{0}^{1} (Ω) \times L^{2} ((- h, 0); L^{2} (Ω))$ which is independent of $ε \in [0, 1]$ , where $\begin{array}{l} (3.22) & B (t) = {(v^{0}, φ) \in Y_{1} : {‖ (v^{0}, φ) ‖}_{Y_{1}} ⩽ R (t), {‖ \frac{d φ}{d s} ‖}_{L^{2} ((- h, 0); L^{2} (Ω))} ⩽ \bar{R} (t)}, \end{array}$ where $\begin{array}{l} ρ_{1} (t) = C (1 + e^{σ t} \int_{- \infty}^{t} e^{σ s} {‖ g (s) ‖}^{2} d s), \\ R^{2} (t) = (1 + β^{- 1} e^{σ h}) ρ_{1} (t), R (t) ⩾ 0, \\ {\bar{R}}^{2} (t) = C ρ_{1}^{\frac{N}{N - 2}} (t) + C ρ_{1} (t) + C ‖ g ‖_{L^{2} ([t - h, t]; L^{2} (Ω))}^{2} + C, \bar{R} (t) ⩾ 0 . \end{array}$

Proof.

First, we observe that for all $t \in R$ , $\begin{array}{l} (3.23) & B (t) \subset {(v^{0}, φ) \in Y : {‖ (v^{0}, φ) ‖}_{Y} ⩽ R (t)}, \end{array}$ with $\begin{array}{l} (3.24) & lim_{t \to - \infty} e^{σ t} R^{\frac{2 N}{N - 2}} (t) = 0, \end{array}$ and so $\hat{B} \in D$ .

Then we are now concerned with the asymptotic estimate using $R (t)$ for fixed $t \in R$ . It may be proved as follows. By definition, for any $ε \in [0, 1]$ , we have $\begin{array}{l} (3.25) & {‖ U_{ε} (t, τ) (u^{0}, ϕ) ‖}_{Y_{1}}^{2} = {‖ \nabla u (t) ‖}^{2} + \int_{t - h}^{t} {‖ \nabla u (s) ‖}^{2} d s . \end{array}$ From (3.2), for any $t - h ⩾ τ$ , subsitituting (3.1) and (3.2) in (3.25), using the definition of $ρ_{1} (t)$ , we obtain $\begin{matrix} {‖ U_{ε} (t, τ) (u^{0}, ϕ) ‖}_{Y_{1}}^{2} \\ ⩽ C β^{- 1} e^{- σ (t - τ - h)} {‖ (u^{0}, ϕ) ‖}_{Y}^{\frac{2 N}{N - 2}} (1 + β^{- 1} e^{σ h}) + (1 + β^{- 1} e^{σ h}) ρ_{1} (t) \\ ⩽ C β^{- 1} e^{- σ (t - τ - h)} {‖ (u^{0}, ϕ) ‖}_{Y}^{\frac{2 N}{N - 2}} (1 + β^{- 1} e^{σ h}) + R^{2} (t), \end{matrix}$ for all $t - h ⩾ τ$ and all $(u^{0}, ϕ) \in Y$ . Hence, because of $e^{σ τ} \to 0$ when $τ \to - \infty$ , we find that $\begin{array}{l} (3.26) & {‖ U_{ε} (t, τ) (u^{0}, ϕ) ‖}_{Y_{1}}^{2} ⩽ R^{2} (t) . \end{array}$

Next, we consider the asymptotic estimate using $\bar{R} (t)$ . We assume that $t - 2 h ⩾ τ$ . Multiplying (1.1) by $\partial_{t} u$ and integrating it over Ω, we arrive at $\begin{array}{l} ‖ \partial_{t} u ‖^{2} + ε ‖ \nabla \partial_{t} u ‖^{2} + \frac{1}{2} \frac{d}{d t} ‖ \nabla u ‖^{2} + \frac{d}{d t} \int_{Ω} F (u) d x = ⟨ b (t, u_{t}), \partial_{t} u ⟩ + ⟨ g (t), \partial_{t} u ⟩ . \end{array}$ By the Cauchy and Young inequalities, we observe that $\begin{array}{l} (3.27) & \frac{1}{2} ‖ \partial_{t} u ‖^{2} + \frac{1}{2} \frac{d}{d t} (‖ \nabla u ‖^{2} + \int_{Ω} F (u) d x) ⩽ {‖ b (t, u_{t}) ‖}^{2} + {‖ g (t) ‖}^{2} . \end{array}$ Integrating (3.27) over $[t - h, t]$ , we notice that $\begin{matrix} \int_{t - h}^{t} {‖ \frac{d}{d t} u (s) ‖}^{2} d s + ({‖ \nabla u (t) ‖}^{2} + 2 \int_{Ω} F (u (t)) d x) \\ ⩽ {‖ \nabla u (t - h) ‖}^{2} + 2 \int_{Ω} F (u (t - h)) d x + 2 \int_{t - h}^{t} {‖ b (s, u_{s}) ‖}^{2} d s + 2 \int_{t - h}^{t} {‖ g (s) ‖}^{2} d s . \end{matrix}$ From (II), (IV) and the Poincaré’s inequality, it follows $\begin{matrix} \int_{t - h}^{t} {‖ \frac{d}{d t} u (s) ‖}^{2} d s \\ ⩽ {‖ \nabla u (t - h) ‖}^{2} + 2 \int_{Ω} F (u (t - h)) d x + \frac{2 C_{b}}{λ_{1}} \int_{t - 2 h}^{t} {‖ \nabla u (s) ‖}^{2} d s + 2 \int_{t - h}^{t} {‖ g (s) ‖}^{2} d s . \end{matrix}$ By (3.13), the Young inequality and the fact that $2 < \frac{2 N}{N - 2}$ , we observe that $\begin{array}{l} \int_{t - h}^{t} {‖ \frac{d}{d t} u (s) ‖}^{2} d s ⩽ C ({‖ \nabla u (t - h) ‖}^{\frac{2 N}{N - 2}} + 1 + ‖ g ‖_{L^{2} ([t - h, t]; L^{2} (Ω))}^{2} + \int_{t - 2 h}^{t} {‖ \nabla u (s) ‖}^{2} d s) . \end{array}$ Now, we estimate $‖ \nabla u (t - h) ‖^{2}$ . Replacing t by $t - h$ in (3.1), we obtain $\begin{array}{l} {‖ \nabla u (t - h) ‖}^{2} ⩽ C e^{- σ (t - h - τ)} {‖ (u^{0}, ϕ) ‖}_{Y}^{\frac{2 N}{N - 2}} + C (1 + e^{- σ (t - h)} \int_{- \infty}^{t - h} e^{σ s} {‖ g (s) ‖}^{2} d s) . \end{array}$ Because $t - h ⩽ t$ and $e^{σ h} > 1$ , we have $\begin{matrix} {‖ \nabla u (t - h) ‖}^{2} & ⩽ C e^{- σ (t - h - τ)} {‖ (u^{0}, ϕ) ‖}_{Y}^{\frac{2 N}{N - 2}} + C e^{σ h} (1 + e^{- σ t} \int_{- \infty}^{t} e^{σ s} {‖ g (s) ‖}^{2} d s) \\ ⩽ C e^{σ h} e^{- σ (t - τ)} {‖ (u^{0}, ϕ) ‖}_{Y}^{\frac{2 N}{N - 2}} + e^{σ h} ρ_{1} (t) . \end{matrix}$ Hence $\begin{array}{l} {‖ \nabla u (t - h) ‖}^{\frac{2 N}{N - 2}} ⩽ {(C e^{σ h} e^{- σ (t - τ)} {‖ (u^{0}, ϕ) ‖}_{Y}^{\frac{2 N}{N - 2}} + e^{σ h} ρ_{1} (t))}^{\frac{N}{N - 2}} . \end{array}$ Since $1 < \frac{2 N}{N - 2}$ , using a convexity argument, we derive $\begin{array}{l} (3.28) & {‖ \nabla u (t - h) ‖}^{\frac{2 N}{N - 2}} ⩽ C e^{- \frac{N}{N - 2} σ (t - τ)} {‖ (u^{0}, ϕ) ‖}_{Y}^{\frac{2 N^{2}}{{(N - 2)}^{2}}} + e^{σ h} ρ_{1}^{\frac{N}{N - 2}} (t) . \end{array}$ Taking $2 h$ in place of h in (3.2), we get that for any $τ ⩽ t - 2 h$ , $\begin{array}{l} (3.29) & \int_{t - 2 h}^{t} {‖ \nabla u (s) ‖}^{2} d s ⩽ C e^{- σ (t - τ)} {‖ (u^{0}, ϕ) ‖}_{Y}^{\frac{2 N}{N - 2}} + C ρ_{1} (t) . \end{array}$ From (3.28) and (3.29), it follows that for all $τ ⩽ t - 2 h$ and all $(u^{0}, ϕ) \in Y$ , $\begin{matrix} \int_{t - h}^{t} {‖ \frac{d}{d t} u (s) ‖}^{2} d s ⩽ & C e^{- \frac{N}{N - 2} σ (t - τ)} {‖ (u^{0}, ϕ) ‖}_{Y}^{\frac{2 N^{2}}{{(N - 2)}^{2}}} + C e^{- σ (t - τ)} {‖ (u^{0}, ϕ) ‖}_{Y}^{\frac{2 N}{N - 2}} \\ + C ρ_{1}^{\frac{2 N}{N - 2}} (t) + C ρ_{1} (t) + C ‖ g ‖_{L^{2} ([t - h, t]; L^{2} (Ω))}^{2} + C . \end{matrix}$ Hence, for all $τ ⩽ t - 2 h$ and for any $(u^{0}, ϕ) \in Y$ , we deduce that $\begin{array}{l} \int_{t - h}^{t} {‖ \frac{d}{d t} u (s) ‖}^{2} d s ⩽ C e^{- \frac{N}{N - 2} σ (t - τ)} {‖ (u^{0}, ϕ) ‖}_{Y}^{\frac{2 N^{2}}{{(N - 2)}^{2}}} + C e^{- σ (t - τ)} {‖ (u^{0}, ϕ) ‖}_{Y}^{\frac{2 N}{N - 2}} + {\bar{R}}^{2} (t) . \end{array}$ Because of $e^{σ τ} \to 0$ when $τ \to - \infty$ , we obtain $\begin{array}{l} (3.30) & \int_{t - h}^{t} {‖ \frac{d}{d t} u (s) ‖}^{2} d s ⩽ {\bar{R}}^{2} (t) . \end{array}$ Consequently, combining (3.26) with (3.30), we know that there exists a pullback $D$ -absorbing family $\hat{B} = {B (t)}_{t \in R}$ in $Y = H_{0}^{1} (Ω) \times L^{2} ((- h, 0); L^{2} (Ω))$ which is independent of $ε \in [0, 1]$ . □

Lemma 3.5.

Assume that $(H 1)$ – $(H 3)$ are satisfied. Then for any $τ + h < a$ , $[a, b] \subset R$ , and bounded set $D \subset Y = H_{0}^{1} (Ω) \times L^{2} ((- h, 0); L^{2} (Ω))$ , $\begin{array}{l} (3.31) & sup_{\begin{array}{c} t \in [a, b] \\ (u^{0}, ϕ) \in D \end{array}} {‖ U_{ε} (t, τ) (u^{0}, ϕ) - U_{ε_{0}} (t, τ) (u^{0}, ϕ) ‖}_{Y_{1}} \overset{ε \to ε_{0}}{⟶} 0, for any ε_{0} \in [0, 1] . \end{array}$

Proof.

Let $(u (t), u_{t}) = U_{ε} (t, τ) (u^{0}, ϕ)$ with initial data $(u (τ), u_{τ}) = (u^{0}, ϕ)$ , where $u (t)$ is the solution to (1.1). And let $(\bar{u} (t), {\bar{u}}_{t}) = U_{ε_{0}} (t, τ) (u^{0}, ϕ)$ with $ε = ε_{0}$ and initial data $(\bar{u} (τ), {\bar{u}}_{τ}) = (u^{0}, ϕ)$ where $\bar{u} (t)$ is the solution to (1.1).

Setting $(u (t), u_{t}) - (\bar{u} (t), {\bar{u}}_{t}) = (u (t) - \bar{u} (t), u_{t} - {\bar{u}}_{t})$ , $w (t) = u (t) - \bar{u} (t)$ and $w_{t} = u_{t} - {\bar{u}}_{t}$ , then $w (τ) = 0$ and $w (τ + θ, x) = ϕ (θ, x) - ϕ (θ, x) = 0$ , $θ \in [- h, 0]$ . Hence, the following equality holds true $\begin{array}{l} (3.32) & \partial_{t} w - Δ w - ε_{0} Δ \partial_{t} w - (ε - ε_{0}) Δ \partial_{t} u + f (u) - f (\bar{u}) = b (t, u_{t}) - b (t, {\bar{u}}_{t}) . \end{array}$ Multiplying (3.32) by w and integrating it over Ω, we have $\begin{array}{l} (3.33) & \begin{matrix} \frac{1}{2} \frac{d}{d t} {‖ w (t) ‖}^{2} + {‖ \nabla w (t) ‖}^{2} + \frac{1}{2} ε_{0} \frac{d}{d t} {‖ \nabla w (t) ‖}^{2} + (ε - ε_{0}) ⟨ - Δ \partial_{t} u, w ⟩ + ⟨ f (u) - f (\bar{u}), w ⟩ \\ = ⟨ b (t, u_{t}) - b (t, {\bar{u}}_{t}), w ⟩ . \end{matrix} \end{array}$ By Young inequality and (1.3), we derive that $\begin{array}{l} (3.34) & | (ε - ε_{0}) ⟨ - Δ \partial_{t} u, w ⟩ | ⩽ \frac{{(ε - ε_{0})}^{2}}{2} {‖ \nabla \partial_{t} u (t) ‖}^{2} + \frac{1}{2} {‖ \nabla w (t) ‖}^{2}, \\ (3.35) & ⟨ f (u) - f (\bar{u}), w ⟩ ⩾ - l {‖ w (t) ‖}^{2}, \\ (3.36) & ⟨ b (t, u_{t}) - b (t, {\bar{u}}_{t}), w ⟩ ⩽ \frac{1}{2} {‖ b (t, u_{t}) - b (t, {\bar{u}}_{t}) ‖}^{2} + \frac{1}{2} {‖ w (t) ‖}^{2} . \end{array}$ From (3.33)–(3.36) and (III), it follows that $\begin{array}{l} (3.37) & \begin{matrix} \frac{d}{d t} ({‖ w (t) ‖}^{2} + ε_{0} {‖ \nabla w (t) ‖}^{2}) ⩽ & (2 l + 1) ({‖ w (t) ‖}^{2} + ε_{0} {‖ \nabla w (t) ‖}^{2}) \\ + {(ε - ε_{0})}^{2} ‖ \nabla \partial_{t} u ‖^{2} + L_{b} ‖ u_{t} - {\bar{u}}_{t} ‖_{L^{2} ((- h, 0); L^{2} (Ω))}^{2} . \end{matrix} \end{array}$ Integrating (3.37) over $[τ, t]$ , we obtain $\begin{matrix} {‖ w (t) ‖}^{2} + ε_{0} {‖ \nabla w (t) ‖}^{2} - ({‖ w (τ) ‖}^{2} + ε_{0} {‖ \nabla w (τ) ‖}^{2}) \\ ⩽ {(ε - ε_{0})}^{2} \int_{τ}^{t} {‖ \nabla \partial_{t} u (s) ‖}^{2} d s + (2 l + 1) \int_{τ}^{t} ({‖ w (s) ‖}^{2} + ε_{0} {‖ \nabla w (s) ‖}^{2}) d s \\ + L_{b} \int_{τ}^{t} \int_{- h}^{0} {‖ w (s + θ) ‖}^{2} d θ d s \\ ⩽ {(ε - ε_{0})}^{2} \int_{τ}^{t} {‖ \nabla \partial_{t} u (s) ‖}^{2} d s + (2 l + 1) \int_{τ}^{t} ({‖ w (s) ‖}^{2} + ε_{0} {‖ \nabla w (s) ‖}^{2}) d s \\ + L_{b} h \int_{τ - h}^{τ} {‖ w (s) ‖}^{2} d s + L_{b} h \int_{τ}^{t} {‖ w (s) ‖}^{2} d s . \end{matrix}$ Taking $α > 2 l + 1 + L_{b} h$ , we obtain $\begin{array}{l} (3.38) & \begin{matrix} {‖ w (t) ‖}^{2} + ε_{0} {‖ \nabla w (t) ‖}^{2} ⩽ & {‖ w (τ) ‖}^{2} + ε_{0} {‖ \nabla w (τ) ‖}^{2} + {(ε - ε_{0})}^{2} \int_{τ}^{t} {‖ \nabla \partial_{t} u (s) ‖}^{2} d s \\ + α \int_{τ}^{t} ({‖ w (s) ‖}^{2} + ε_{0} {‖ \nabla w (s) ‖}^{2}) d s + L_{b} h \int_{τ - h}^{τ} {‖ w (s) ‖}^{2} d s . \end{matrix} \end{array}$ Applying the Gronwall lemma to (3.38), we notice that $\begin{array}{l} {‖ w (t) ‖}^{2} + ε_{0} {‖ \nabla w (t) ‖}^{2} \\ ⩽ ({‖ w (τ) ‖}^{2} + ε_{0} {‖ \nabla w (τ) ‖}^{2} + {(ε - ε_{0})}^{2} \int_{τ}^{t} {‖ \nabla \partial_{t} u (s) ‖}^{2} d s \\ (3.39) & + L_{b} h \int_{- h}^{0} {‖ w (τ + θ) ‖}^{2} d θ) e^{α (t - τ)} . \end{array}$ Since $w (τ) = 0$ and $w (τ + θ, x) = ϕ (θ, x) - ϕ (θ, x) = 0$ , $θ \in [- h, 0]$ , the estimate (3.39) shows that $\begin{array}{l} (3.40) & {‖ w (t) ‖}^{2} + ε_{0} {‖ \nabla w (t) ‖}^{2} ⩽ {(ε - ε_{0})}^{2} e^{α (t - τ)} \int_{τ}^{t} {‖ \nabla \partial_{t} u (s) ‖}^{2} d s . \end{array}$ Now we divide the agrument into two cases.

Case 1: $ε_{0} > 0$ . From Lemma 3.2 and (3.40), it is readily seen that $\begin{array}{l} (3.41) & lim_{ε \to ε_{0}} sup_{\begin{array}{c} t \in [a, b] \\ (u^{0}, ϕ) \in D \end{array}} {‖ \nabla w (t) ‖}^{2} ⩽ lim_{ε \to ε_{0}} sup_{\begin{array}{c} t \in [a, b] \\ (u^{0}, ϕ) \in D \end{array}} \frac{{(ε - ε_{0})}^{2}}{ε_{0} ε} e^{α (t - τ)} \cdot ε \int_{τ}^{t} {‖ \nabla \partial_{t} u (s) ‖}^{2} d s = 0 . \end{array}$ Case 2: $ε_{0} = 0$ . We simplify (3.40) as follows $\begin{array}{l} (3.42) & {‖ w (t) ‖}^{2} ⩽ ε^{2} e^{α (t - τ)} \int_{τ}^{t} {‖ \nabla \partial_{t} u (s) ‖}^{2} d s, \end{array}$ and $\begin{array}{l} (3.43) & lim_{ε \to 0} sup_{\begin{array}{c} t \in [a, b] \\ (u^{0}, ϕ) \in D \end{array}} {‖ w (t) ‖}^{2} ⩽ lim_{ε \to ε_{0}} sup_{\begin{array}{c} t \in [a, b] \\ (u^{0}, ϕ) \in D \end{array}} ε e^{α (t - τ)} \cdot ε \int_{τ}^{t} {‖ \nabla \partial_{t} u (s) ‖}^{2} d s = 0 . \end{array}$ Multiplying (3.32) by w and using (3.35), we find $\begin{array}{l} (3.44) & \begin{matrix} {‖ \nabla w (t) ‖}^{2} ⩽ & ‖ \partial_{t} w (t) ‖ ‖ w (t) ‖ + \frac{ε}{2} {‖ \nabla \partial_{t} u (t) ‖}^{2} + \frac{1}{2} {‖ \nabla w (t) ‖}^{2} \\ + l {‖ w (t) ‖}^{2} + ‖ b (t, u_{t}) - b (t, {\bar{u}}_{t}) ‖ ‖ w (t) ‖ . \end{matrix} \end{array}$ By using (I) and (III), we get $\begin{array}{l} (3.45) & \begin{matrix} {‖ \nabla w (t) ‖}^{2} ⩽ & C (‖ \partial_{t} w (t) ‖ ‖ w (t) ‖ + ε {‖ \nabla \partial_{t} u (t) ‖}^{2} + {‖ w (t) ‖}^{2} \\ + ‖ u_{t} - {\bar{u}}_{t} ‖_{L^{2} ((- h, 0); L^{2} (Ω))} ‖ w (t) ‖), \end{matrix} \end{array}$ which, together with (I), Lemmas 3.2–3.3, inequality (3.41) and (3.43), gives $\begin{array}{l} (3.46) & lim_{ε \to ε_{0}} sup_{\begin{array}{c} t \in [a, b] \\ (u^{0}, ϕ) \in D \end{array}} {‖ \nabla w (t) ‖}^{2} = 0 . \end{array}$ for any $ε_{0} \in [0, 1]$ , $t \in [a, b]$ and $(u^{0}, ϕ) \in D$ .

Then we will estimate $\int_{t - h}^{t} ‖ \nabla w (s) ‖^{2} d s$ . Using (3.33), the Cauchy inequality, Young inequality and (1.3), we find that $\begin{array}{l} (3.47) & \begin{matrix} \frac{d}{d t} ({‖ w (t) ‖}^{2} + ε_{0} {‖ \nabla w (t) ‖}^{2}) + (2 - ε_{1} - \frac{2 l + ε_{2}}{λ_{1}}) {‖ \nabla w (t) ‖}^{2} \\ ⩽ \frac{1}{ε_{1}} {(ε - ε_{0})}^{2} {‖ \nabla \partial_{t} u (t) ‖}^{2} + \frac{1}{ε_{2}} {‖ b (t, u_{t}) - b (t, {\bar{u}}_{t}) ‖}^{2}, \end{matrix} \end{array}$ where $ε_{1}$ and $ε_{2}$ are positive constants. Multiplying (3.47) by $e^{σ t}$ , it follows that $\begin{array}{l} (3.48) & \begin{matrix} \frac{d}{d t} (e^{σ t} ({‖ w (t) ‖}^{2} + ε_{0} {‖ \nabla w (t) ‖}^{2})) + (2 - ε_{1} - \frac{2 l + ε_{2}}{λ_{1}}) e^{σ t} {‖ \nabla w (t) ‖}^{2} \\ ⩽ \frac{1}{ε_{1}} {(ε - ε_{0})}^{2} e^{σ t} {‖ \nabla \partial_{t} u (t) ‖}^{2} + \frac{1}{ε_{2}} e^{σ t} {‖ b (t, u_{t}) - b (t, {\bar{u}}_{t}) ‖}^{2} \\ + σ e^{σ t} ({‖ w (t) ‖}^{2} + ε_{0} {‖ \nabla w (t) ‖}^{2}) . \end{matrix} \end{array}$ Integrating (3.48) from τ to t, we have $\begin{array}{l} (3.49) & \begin{matrix} {‖ w (t) ‖}^{2} + ε_{0} {‖ \nabla w (t) ‖}^{2} + (2 - ε_{1} - \frac{2 l + ε_{2}}{λ_{1}}) e^{- σ t} \int_{τ}^{t} e^{σ s} {‖ \nabla w (s) ‖}^{2} d s \\ ⩽ e^{- σ (t - τ)} ({‖ w (τ) ‖}^{2} + ε_{0} {‖ \nabla w (τ) ‖}^{2}) + σ e^{- σ t} \int_{τ}^{t} e^{σ s} ({‖ w (s) ‖}^{2} + ε_{0} {‖ \nabla w (s) ‖}^{2}) d s \\ + \frac{1}{ε_{1}} {(ε - ε_{0})}^{2} e^{- σ t} \int_{τ}^{t} e^{σ s} {‖ \nabla \partial_{t} u (s) ‖}^{2} d s + \frac{1}{ε_{2}} e^{- σ t} \int_{τ}^{t} e^{σ s} {‖ b (s, u_{s}) - b (s, {\bar{u}}_{s}) ‖}^{2} d s . \end{matrix} \end{array}$ Thanks to (V) and the assumptions of $w (t)$ , we observe that $\begin{array}{l} (3.50) & e^{- σ (t - τ)} ({‖ w (τ) ‖}^{2} + ε_{0} {‖ \nabla w (τ) ‖}^{2}) = 0, \\ (3.51) & σ e^{- σ t} \int_{τ}^{t} e^{σ s} ({‖ w (s) ‖}^{2} + ε_{0} {‖ \nabla w (s) ‖}^{2}) d s ⩽ \frac{(1 + λ_{1}) σ}{λ_{1}} e^{- σ t} \int_{τ}^{t} e^{σ s} {‖ \nabla w (s) ‖}^{2} d s, \\ (3.52) & \begin{matrix} \frac{1}{ε_{2}} e^{- σ t} \int_{τ}^{t} e^{σ s} {‖ b (s, u_{s}) - b (s, {\bar{u}}_{s}) ‖}^{2} d s \\ ⩽ \frac{C_{b}}{ε_{2}} e^{- σ t} \int_{τ}^{t} e^{σ s} {‖ w (s) ‖}^{2} d s + \frac{C_{b}}{ε_{2}} e^{- σ t} \int_{τ - h}^{τ} e^{σ s} {‖ w (s) ‖}^{2} d s \\ ⩽ \frac{C_{b}}{ε_{2} \cdot λ_{1}} e^{- σ t} \int_{τ}^{t} e^{σ s} {‖ \nabla w (s) ‖}^{2} d s + \frac{C_{b}}{ε_{2}} e^{- σ t} \int_{- h}^{0} e^{σ s} {‖ w (τ + θ) ‖}^{2} d θ \\ ⩽ \frac{C_{b}}{ε_{2} \cdot λ_{1}} e^{- σ t} \int_{τ}^{t} e^{σ s} {‖ \nabla w (s) ‖}^{2} d s . \end{matrix} \end{array}$ From (3.49)–(3.52), it follows $\begin{array}{l} (3.53) & \begin{matrix} (2 - ε_{1} - \frac{2 l + ε_{2}}{λ_{1}} - \frac{(1 + λ_{1}) σ}{λ_{1}} - \frac{C_{b}}{ε_{2} \cdot λ_{1}}) e^{- σ t} \int_{τ}^{t} e^{σ s} {‖ \nabla w (s) ‖}^{2} d s \\ ⩽ \frac{1}{ε_{1}} {(ε - ε_{0})}^{2} e^{- σ t} \int_{τ}^{t} e^{σ s} {‖ \nabla \partial_{t} u (s) ‖}^{2} d s . \end{matrix} \end{array}$ Furthermore, for $τ ⩽ t - h$ , we have, as $[t - h, t] \subset [τ, t]$ , $\begin{array}{l} (3.54) & \int_{τ}^{t} e^{σ s} {‖ \nabla w (s) ‖}^{2} d s ⩾ \int_{t - h}^{t} e^{σ s} {‖ \nabla w (s) ‖}^{2} d s ⩾ e^{σ (t - h)} \int_{t - h}^{t} {‖ \nabla w (s) ‖}^{2} d s . \end{array}$ Hence, we obtain from (3.53)–(3.54) $\begin{array}{l} (3.55) & \begin{matrix} (2 - ε_{1} - \frac{2 l + ε_{2}}{λ_{1}} - \frac{(1 + λ_{1}) σ}{λ_{1}} - \frac{C_{b}}{ε_{2} \cdot λ_{1}}) \int_{t - h}^{t} {‖ \nabla w (s) ‖}^{2} d s \\ ⩽ \frac{1}{ε_{1}} \frac{{(ε - ε_{0})}^{2}}{ε} e^{- σ (t - h)} \cdot ε \int_{τ}^{t} e^{σ s} {‖ \nabla \partial_{t} u (s) ‖}^{2} d s . \end{matrix} \end{array}$ From Lemma 3.1, we can choose $ε_{1}$ and σ small enough to satisfy $\begin{array}{l} (3.56) & β_{1} : = (2 - \frac{2 l + ε_{2} + \frac{C_{b}}{ε_{2}}}{λ_{1}} - ε_{1} - \frac{(1 + λ_{1}) σ}{λ_{1}}) > 0 . \end{array}$ Then it follows that $\begin{array}{l} (3.57) & \begin{matrix} lim_{ε \to ε_{0}} sup_{\begin{array}{c} t \in [a, b] \\ (u^{0}, ϕ) \in D \end{array}} \int_{t - h}^{t} {‖ \nabla w (s) ‖}^{2} d s \\ ⩽ lim_{ε \to ε_{0}} sup_{\begin{array}{c} t \in [a, b] \\ (u^{0}, ϕ) \in D \end{array}} \frac{1}{ε_{1} \cdot β_{1}} \frac{{(ε - ε_{0})}^{2}}{ε} e^{- σ (t - h)} \cdot ε \int_{τ}^{t} e^{σ s} {‖ \nabla \partial_{t} u (s) ‖}^{2} d s \\ = 0 . \end{matrix} \end{array}$ Consequently, due (3.46) and (3.57), it is readily seen that for any $ε_{0} \in [0, 1]$ when $t ⩾ τ + h$ , $\begin{array}{l} (3.58) & \begin{matrix} lim_{ε \to ε_{0}} sup_{\begin{array}{c} t \in [a, b] \\ (u^{0}, ϕ) \in D \end{array}} {‖ U_{ε} (t, τ) (u^{0}, ϕ) - U_{ε_{0}} (t, τ) (u^{0}, ϕ) ‖}_{Y_{1}} \\ = lim_{ε \to ε_{0}} sup_{\begin{array}{c} t \in [a, b] \\ (u^{0}, ϕ) \in D \end{array}} ({‖ \nabla w (t) ‖}^{2} + \int_{t - h}^{t} {‖ \nabla w (s) ‖}^{2} d s) \\ = 0 . \end{matrix} \end{array}$ The proof is hence finished. □

Lemma 3.6.

Under assumptions $(H 1)$ – $(H 3)$ , for any $τ + h < a$ , $[a, b] \in R$ , all sequences ${(u_{n}^{0}, ϕ_{n})}_{n \in N} \subset Y = H_{0}^{1} (Ω) \times L^{2} ((- h, 0); L^{2} (Ω))$ with $(u_{n}^{0}, ϕ_{n}) \overset{n \to \infty}{⟶} (u^{0}, ϕ)$ in $Y$ and ${t_{n}}_{n \in N} \subset [a, b]$ with $t_{n} \to t_{0}$ , $\begin{array}{l} (3.59) & {‖ U_{ε} (t_{n}, τ) (u_{n}^{0}, ϕ_{n}) - U_{ε} (t_{0}, τ) (u^{0}, ϕ) ‖}_{Y_{1}} \overset{n \to \infty}{⟶} 0, for any ε \in [0, 1] . \end{array}$

Proof.

Let $(u_{n} (t), u_{n t}) = U_{ε} (t, τ) (u_{n}^{0}, ϕ_{n})$ with initial data $(u_{n} (τ), u_{n τ}) = (u_{n}^{0}, ϕ_{n})$ , where $u_{n} (t)$ is the solution to (1.1), and let $(u_{0} (t), u_{0 t}) = U_{ε} (t, τ) (u^{0}, ϕ)$ with initial data $(u_{0} (τ), u_{0 τ}) = (u^{0}, ϕ)$ where $u_{0} (t)$ is the solution to (1.1).

Setting $(u_{n} (t), u_{n t}) - (u_{0} (t), u_{0 t}) = (u_{n} (t) - u_{0} (t), u_{n t} - u_{0 t})$ , $w_{n} (t) = u_{n} (t) - u_{0} (t)$ and $w_{n t} = u_{n t} - u_{0 t}$ , then $w_{n} (τ) = u_{n}^{0} - u^{0}$ and $w_{n} (τ + θ, x) = ϕ_{n} (θ, x) - ϕ_{n} (θ, x) = 0$ , $θ \in [- h, 0]$ . Hence, the following equation holds true $\begin{array}{l} (3.60) & \partial_{t} w_{n} - ε Δ \partial_{t} w_{n} - Δ w_{n} + f (u_{n}) - f (u_{0}) = b (t, u_{n t}) - b (t, u_{0 t}) . \end{array}$ Multiplying (3.60) by $w_{n}$ and integrating over Ω, we arrive at $\begin{array}{l} (3.61) & \begin{matrix} \frac{1}{2} \frac{d}{d t} {‖ w_{n} (t) ‖}^{2} + \frac{1}{2} ε \frac{d}{d t} {‖ \nabla w_{n} (t) ‖}^{2} + {‖ \nabla w_{n} (t) ‖}^{2} + ⟨ f (u_{n}) - f (u_{0}), w_{n} ⟩ \\ = ⟨ b (t, u_{n t}) - b (t, u_{0 t}), w_{n} ⟩ . \end{matrix} \end{array}$ By the Young inequality and (1.3), we derive that $\begin{array}{l} (3.62) & ⟨ f (u_{n}) - f (u_{0}), w_{n} ⟩ ⩾ - l {‖ w_{n} (t) ‖}^{2}, \\ (3.63) & ⟨ b (t, u_{n t}) - b (t, u_{0 t}), w_{n} ⟩ ⩽ \frac{1}{2} {‖ b (t, u_{n t}) - b (t, u_{0 t}) ‖}^{2} + \frac{1}{2} {‖ w_{n} (t) ‖}^{2} . \end{array}$ From (3.61)–(3.63) and (III), it follows that $\begin{matrix} \frac{d}{d t} ({‖ w_{n} (t) ‖}^{2} + ε {‖ \nabla w_{n} (t) ‖}^{2}) \\ ⩽ (2 l + 1) ({‖ w_{n} (t) ‖}^{2} + ε {‖ \nabla w_{n} (t) ‖}^{2}) + L_{b} ‖ u_{n t} - u_{0 t} ‖_{L^{2} ((- h, 0); L^{2} (Ω))}^{2}, \end{matrix}$ which, integrated over $[τ, t]$ , gives $\begin{matrix} {‖ w_{n} (t) ‖}^{2} + ε {‖ \nabla w_{n} (t) ‖}^{2} - ({‖ w_{n} (τ) ‖}^{2} + ε {‖ \nabla w_{n} (τ) ‖}^{2}) \\ ⩽ (2 l + 1) \int_{τ}^{t} ({‖ w_{n} (s) ‖}^{2} + ε {‖ \nabla w_{n} (s) ‖}^{2}) d s + L_{b} \int_{τ}^{t} \int_{- h}^{0} {‖ w_{n} (s + θ) ‖}^{2} d θ d s \\ ⩽ (2 l + 1) \int_{τ}^{t} {‖ w_{n} (s) ‖}^{2} d s + L_{b} h \int_{τ - h}^{τ} {‖ w_{n} (s) ‖}^{2} d s + L_{b} h \int_{τ}^{t} {‖ w_{n} (s) ‖}^{2} d s . \end{matrix}$ Taking $α > 2 l + 1 + L_{b} h$ , we get $\begin{array}{l} (3.64) & \begin{matrix} {‖ w_{n} (t) ‖}^{2} + ε {‖ \nabla w_{n} (t) ‖}^{2} \\ ⩽ {‖ w_{n} (τ) ‖}^{2} + ε {‖ \nabla w_{n} (τ) ‖}^{2} + α \int_{τ}^{t} {‖ w_{n} (s) ‖}^{2} d s + L_{b} h \int_{- h}^{0} {‖ w_{n} (τ + θ) ‖}^{2} d θ . \end{matrix} \end{array}$ Applying the Gronwall lemma to (3.64), we conclude that $\begin{array}{l} (3.65) & \begin{matrix} {‖ w_{n} (t) ‖}^{2} + ε {‖ \nabla w_{n} (t) ‖}^{2} \\ ⩽ ({‖ w_{n} (τ) ‖}^{2} + ε {‖ \nabla w_{n} (τ) ‖}^{2} + L_{b} h \int_{- h}^{0} {‖ w_{n} (τ + θ) ‖}^{2} d θ) e^{α (t - τ)} . \end{matrix} \end{array}$ Since $\begin{array}{l} (3.66) & lim_{n \to \infty} {‖ \nabla w_{n} (τ) ‖}^{2} = lim_{n \to \infty} {‖ \nabla (u_{n}^{0} - u^{0}) ‖}^{2} = 0 \end{array}$ and $\begin{array}{l} (3.67) & lim_{n \to \infty} \int_{- h}^{0} {‖ w_{n} (τ + θ) ‖}^{2} d θ = lim_{n \to \infty} {‖ ϕ_{n} (θ) - ϕ (θ) ‖}^{2} d θ = 0, \end{array}$ we can divide the argument into two cases.

Case 1: $ε > 0$ . We readily obtain $\begin{array}{l} (3.68) & lim_{n \to \infty} {‖ \nabla w_{n} (t) ‖}^{2} = 0 . \end{array}$

Case 2: $ε = 0$ . Then we get $\begin{array}{l} (3.69) & lim_{n \to \infty} {‖ w_{n} (t) ‖}^{2} = 0 . \end{array}$ Multiplying (3.60) by $w_{n}$ and integrating it over Ω, by Young inequality and (1.3), it follows that $\begin{array}{l} (3.70) & {‖ \nabla w_{n} (t) ‖}^{2} ⩽ l {‖ w_{n} (t) ‖}^{2} + ‖ \partial_{t} w_{n} (t) ‖ ‖ w_{n} (t) ‖ + ‖ b (t, u_{n t}) - b (t, u_{0 t}) ‖ ‖ w_{n} (t) ‖ . \end{array}$ Applying (I), Lemma 3.3 and (3.69) in (3.70), we get $\begin{array}{l} (3.71) & lim_{n \to \infty} {‖ \nabla w_{n} (t) ‖}^{2} = 0 for any ε \in [0, 1] . \end{array}$

Then we will estimate $\int_{t - h}^{t} ‖ \nabla w_{n} (s) ‖^{2} d s$ . Using (3.61), the Cauchy, Young inequalities and (1.3), we find that $\begin{array}{l} (3.72) & \begin{matrix} \frac{d}{d t} ({‖ w_{n} (t) ‖}^{2} + ε {‖ \nabla w_{n} (t) ‖}^{2}) + (2 - \frac{2 l + ε_{1}}{λ_{1}}) {‖ \nabla w_{n} (t) ‖}^{2} \\ ⩽ \frac{1}{ε_{1}} {‖ b (t, u_{n t}) - b (t, u_{0 t}) ‖}^{2}, \end{matrix} \end{array}$ where $ε_{1}$ is a positive constant. Multiplying (3.72) by $e^{σ t}$ , and integrating it from τ to t, we have $\begin{array}{l} (3.73) & \begin{matrix} {‖ w_{n} (t) ‖}^{2} + ε {‖ \nabla w_{n} (t) ‖}^{2} + (2 - \frac{2 l + ε_{1}}{λ_{1}}) e^{- σ t} \int_{τ}^{t} e^{σ s} {‖ \nabla w_{n} (s) ‖}^{2} d s \\ ⩽ e^{- σ (t - τ)} ({‖ w_{n} (τ) ‖}^{2} + ε {‖ \nabla w_{n} (τ) ‖}^{2}) + \frac{1}{ε_{1}} e^{- σ t} \int_{τ}^{t} e^{σ s} {‖ b (s, u_{n s}) - b (s, u_{0 s}) ‖}^{2} d s \\ + σ e^{- σ t} \int_{τ}^{t} e^{σ s} ({‖ w_{n} (t) ‖}^{2} + ε {‖ \nabla w_{n} (t) ‖}^{2}) d s . \end{matrix} \end{array}$ Thanks to (V), we observe that $\begin{array}{l} σ e^{- σ t} \int_{τ}^{t} e^{σ s} ({‖ w_{n} (s) ‖}^{2} + ε {‖ \nabla w_{n} (s) ‖}^{2}) d s \\ (3.74) & ⩽ \frac{(1 + λ_{1}) σ}{λ_{1}} e^{- σ t} \int_{τ}^{t} e^{σ s} {‖ \nabla w_{n} (s) ‖}^{2} d s, \\ (3.75) & \begin{matrix} \frac{1}{ε_{1}} e^{- σ t} \int_{τ}^{t} e^{σ s} {‖ b (s, u_{n s}) - b (s, u_{0 s}) ‖}^{2} d s \\ ⩽ \frac{C_{b}}{ε_{1}} e^{- σ t} \int_{τ}^{t} e^{σ s} {‖ w_{n} (s) ‖}^{2} d s + \frac{C_{b}}{ε_{1}} e^{- σ t} \int_{τ - h}^{τ} e^{σ s} {‖ w_{n} (s) ‖}^{2} d s \\ ⩽ \frac{C_{b}}{ε_{1} \cdot λ_{1}} e^{- σ t} \int_{τ}^{t} e^{σ s} {‖ \nabla w_{n} (s) ‖}^{2} d s + \frac{C_{b}}{ε_{1}} e^{- σ t} \int_{- h}^{0} e^{σ s} {‖ w_{n} (τ + θ) ‖}^{2} d θ . \end{matrix} \end{array}$ From (3.73)–(3.75), it readily follows $\begin{array}{l} (3.76) & \begin{matrix} (2 - \frac{2 l + ε_{1}}{λ_{1}} - \frac{C_{b}}{ε_{1} \cdot λ_{1}} - \frac{(1 + λ_{1}) σ}{λ_{1}}) e^{- σ t} \int_{τ}^{t} e^{σ s} {‖ \nabla w_{n} (s) ‖}^{2} d s \\ ⩽ e^{- σ (t - τ)} ({‖ w_{n} (τ) ‖}^{2} + ε {‖ \nabla w_{n} (τ) ‖}^{2}) + \frac{C_{b}}{ε_{1}} e^{- σ t} \int_{- h}^{0} e^{σ s} {‖ w_{n} (τ + θ) ‖}^{2} d θ . \end{matrix} \end{array}$ Furthermore, for $τ ⩽ t - h$ , we have $\begin{array}{l} (3.77) & \int_{τ}^{t} e^{σ s} {‖ \nabla w_{n} (s) ‖}^{2} d s ⩾ \int_{t - h}^{t} e^{σ s} {‖ \nabla w_{n} (s) ‖}^{2} d s ⩾ e^{σ (t - h)} \int_{t - h}^{t} {‖ \nabla w_{n} (s) ‖}^{2} d s, \end{array}$ as $[t - h, t] \subset [τ, t]$ . Hence, we obtain $\begin{array}{l} (3.78) & \begin{matrix} (2 - \frac{2 l + ε_{1}}{λ_{1}} - \frac{C_{b}}{ε_{1} \cdot λ_{1}} - \frac{(1 + λ_{1}) σ}{λ_{1}}) \int_{t - h}^{t} e^{σ s} {‖ \nabla w_{n} (s) ‖}^{2} d s \\ ⩽ e^{- σ (t - τ - h)} ({‖ w_{n} (τ) ‖}^{2} + ε {‖ \nabla w_{n} (τ) ‖}^{2}) + \frac{C_{b}}{ε_{1}} e^{- σ (t - h)} \int_{- h}^{0} e^{σ s} {‖ w_{n} (τ + θ) ‖}^{2} d θ . \end{matrix} \end{array}$ From Lemma 3.1, we can choose σ small enough to satisfy $\begin{array}{l} (3.79) & β_{2} : = (2 - \frac{2 l + ε_{1} + \frac{C_{b}}{ε_{1}}}{λ_{1}} - \frac{(1 + λ_{1}) σ}{λ_{1}}) > 0 . \end{array}$ Therefore, (3.78) becomes $\begin{array}{l} (3.80) & \begin{matrix} \int_{t - h}^{t} e^{σ s} {‖ \nabla w_{n} (s) ‖}^{2} d s \\ ⩽ \frac{1}{β_{2}} e^{- σ (t - τ - h)} ({‖ w_{n} (τ) ‖}^{2} + ε {‖ \nabla w_{n} (τ) ‖}^{2}) + \frac{1}{β_{2}} \frac{C_{b}}{ε_{1}} e^{- σ (t - h)} \int_{- h}^{0} e^{σ s} {‖ w_{n} (τ + θ) ‖}^{2} d θ . \end{matrix} \end{array}$ By the assumptions on $w_{n} (t)$ , we infer that $\begin{array}{l} (3.81) & lim_{n \to \infty} \int_{t - h}^{t} {‖ \nabla w_{n} (s) ‖}^{2} d s = 0 . \end{array}$ Combining (3.71) and (3.81), we derive $\begin{array}{l} (3.82) & lim_{n \to \infty} {‖ U_{ε} (t, τ) (u_{n}^{0}, ϕ_{n}) - U_{ε} (t, τ) (u^{0}, ϕ) ‖}_{Y_{1}} = 0 for any t \in [a, b] . \end{array}$ Noting that, $U_{ε} (t, τ) (u^{0}, ϕ) \in C ([a, b]; H_{0}^{1} (Ω) \times L^{2} ((- h, 0); H_{0}^{1} (Ω)))$ , and using (3.82), we conclude $\begin{array}{l} (3.83) & \begin{matrix} {‖ U_{ε} (t_{n}, τ) (u_{n}^{0}, ϕ_{n}) - U_{ε} (t_{0}, τ) (u^{0}, ϕ) ‖}_{Y_{1}} \\ ⩽ {‖ U_{ε} (t_{n}, τ) (u_{n}^{0}, ϕ_{n}) - U_{ε} (t_{n}, τ) (u^{0}, ϕ) ‖}_{Y_{1}} \\ + {‖ U_{ε} (t_{n}, τ) (u^{0}, ϕ) - U_{ε} (t_{0}, τ) (u^{0}, ϕ) ‖}_{Y_{1}} \overset{n \to \infty}{⟶} 0, \end{matrix} \end{array}$ which completes the proof. □

Combining Theorem 2.8 and Lemmas 3.4–3.6, we can prove our main result in the following.

Theorem 3.7.

Under assumptions $(H 1)$ – $(H 3)$ , for any $τ + h < a$ , $[a, b] \in R$ , the pullback $D_{ε}$ -attractor $A_{ε} = {A_{ε} (t)}_{t \in R}$ given by Theorem 2.5 satisfies $\begin{array}{l} lim_{ε \to ε_{0}} sup_{t \in [a, b]} {dist}_{H_{0}^{1} (Ω) \times L^{2} ((- h, 0); L^{2} (Ω))} (A_{ε} (t), A_{ε_{0}} (t)) = 0, for any ε_{0} \in [0, 1] \end{array}$ and $\begin{array}{l} ⋃_{t \in [a, b]} ⋃_{ε \in [0, 1]} A_{ε} (t) is precompact in H_{0}^{1} (Ω) \times L^{2} ((- h, 0); L^{2} (Ω)) . \end{array}$

Footnotes

Acknowledgements

This paper was in part supported by the National Natural Science Foundation of China with contract number 12171082 and the Fundamental Research Funds for the Central Universities with contract number 2232022G-13.

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Upper semicontinuity of pullback attractors for nonclassical diffusion equations with delay

Abstract

Keywords

1. Introduction

Definition 2.1 ([11,15]).

Definition 2.2 ([11,14]).

Definition 2.3 ([14,20]).

Definition 2.4 ([11,14]).

Remark 2.1. dist X ( A , B ) denotes the Hausdorff semidistance between A and B which are bounded sets of X , that is, dist X ( A , B ) = sup x ∈ A inf y ∈ B ‖ x − y ‖ X . Theorem 2.5 ([11,14]).

Definition 2.6 ([21]).

Lemma 2.7 ([20]).

Theorem 2.8 ([21]).

3. Upper semicontinuity

Lemma 3.1 ([11]).

Footnotes

Acknowledgements

References

Remark 2.1.
${dist}_{X} (A, B)$ denotes the Hausdorff semidistance between A and B which are bounded sets of $X$ , that is, ${dist}_{X} (A, B) = {sup}_{x \in A} {inf}_{y \in B} ‖ x - y ‖_{X}$ .

Theorem 2.5 ([11,14]).