Dynamics for a plate equation with nonlinear damping on time-dependent space

Abstract

In this paper, we study the long-time behavior of the following plate equation $\begin{matrix} ε (t) u_{t t} + g (u_{t}) + Δ^{2} u + λ u + f (u) = h, \end{matrix}$ where the coefficient ε depends explicitly on time, the nonlinear damping and the nonlinearity both have critical growths.

Keywords

Plate equation nonlinear damping critical nonlinearity time-dependent space

1. Introduction

In this paper, we consider the following plate equation with nonlinear damping and critical nonlinearity: $\begin{matrix} (1.1) & ε (t) u_{t t} + g (u_{t}) + Δ^{2} u + λ u + f (u) = h (x), x \in Ω, \end{matrix}$ with boundary and initial conditions of the form $\begin{matrix} (1.2) & u |_{\partial Ω} = \frac{\partial}{\partial v} u |_{\partial Ω} = 0, u |_{t = τ} = u_{0}, u_{t} |_{t = τ} = u_{1} . \end{matrix}$

Here $Ω \subset R^{n}$ is a bounded domain with a sufficiently smooth boundary, $λ > 0$ , $h (x) \in L^{2} (Ω)$ , the coefficient $ε (t) \in C^{1} (R)$ is a decreasing bounded function and satisfies $\begin{matrix} (1.3) & lim_{t \to + \infty} ε (t) = 0 . \end{matrix}$

In particular, there exists $L > 0$ such that $\begin{matrix} (1.4) & sup_{t \in R} [| ε (t) | + | ε^{'} (t) |] ⩽ L . \end{matrix}$

The function $f \in C^{1} (R)$ satisfies $\begin{array}{l} (1.5) & | f^{'} (s) | ⩽ C_{1} (1 + | s |^{p}), \\ (1.6) & \underset{| s | \to \infty}{lim inf} \frac{f (s)}{s} > - λ_{1}, \end{array}$ where $0 < p < \infty$ if $n ⩽ 4$ , and $0 < p ⩽ \frac{4}{n - 4}$ if $n > 4$ , $λ_{1}$ is the best constant in the $P o i n c \overset{´}{a} r e$ type inequality $\int_{Ω} | \nabla u |^{2} ⩾ λ_{1}^{\frac{1}{2}} \int_{Ω} | u |^{2}$ . The number $\frac{4}{n - 4}$ is called the critical exponent, since the nonlinearity f is not compact in this case. The damping term $g \in C^{1} (R)$ satisfies $\begin{array}{l} (1.7) & g (0) = 0, g is strictly increasing, and \underset{| s | \to \infty}{lim inf} g^{'} (s) > 0, \\ (1.8) & | g (s) | ⩽ C_{2} (1 + | s |^{q}), \end{array}$ where $1 ⩽ q < \infty$ if $n ⩽ 4$ , and $1 ⩽ q ⩽ \frac{n + 4}{n - 4}$ if $n > 4$ .

The plate equation arises in the nonlinear theory of oscillations. When $ε (t)$ is a positive constant function, the equation (1.1) becomes the autonomous plate equation. This problem has been considered extensively. In [9], Khanmamedov established the existence of global attractor for (1.1) and (1.2) with damping $a (x) u_{t}$ and critical nonlinearity, later on, he extended the result with localized damping and critical exponent in [11]. In [16], the authors showed the existence of global attractor for (1.1) with nonlinear damping. Von Karman equation is one of the most important plate model (see [2,4,10,12] for more details), in [1], the authors investigated the plated equation with nonlinear damping and a logarithmic source term.

When $ε (t)$ depends on time(not a constant function), such as a positive decreasing function vanishing at infinity, i.e., the coefficient of the differential operator depend on time explicitly, the natural energy associated with the system reads $\begin{matrix} ε (t) = \int_{Ω} {| Δ u (x, t) |}^{2} d x + λ \int_{Ω} {| u (x, t) |}^{2} d x + ε (t) \int_{Ω} {| u_{t} (x, t) |}^{2} d x, \end{matrix}$ it is easy to see that the vanishing of ε at infinity prevents the existence of a pullback absorbing set in the usual sense. And the classical theory generally fails to capture the dissipation mechanism due to time-dependent terms at a functional level, as mentioned in [7,8,15]. In recent years, this problem has been studied by many authors, see, for example, [7,8,14,15]. In [8], the authors introduced the notion of the time dependent global attractor. Later on, in [7], the authors considered the long-time behavior of the wave equation with the linear damping. Moreover, in [6], the authors discussed a specific one-dimensional wave equation and established the existence of an invariant time-dependent attractor, which converges in a suitable sense to the classical Fourier equation. Recently, in [15], the authors proved the existence of time-dependent global attractor for the wave equation with critical f and subcritical g. In [14], the authors investigated the asymptotic behavior of the non-autonomous Berger equation on time-dependent space. The study of wave equation on time-dependent space has attracted much attention in recent years. However, the plate equation on time-dependent space is less discussed, especially for the critical nonlinear damping and critical nonlinearity.

The main aim of this paper is to consider the long time behavior of the solution of the problem (1.1) on time-dependent space, where $ε (t)$ depends on time explicitly, the nonlinear damping g and the nonlinearity f both have critical exponents. The difficulty for verifying compactness mainly comes from the critical nonlinear damping and the critical nonlinear term.

This paper is organized as follows: in Section 2, we give some preparations for our consideration; in Section 3, we establish some dissipativity of the process by using appropriate energy estimates; in Section 4, we derive the compactness of the process, and obtain the existence of time-dependent global attractor.

2. Preliminaries

In this section, firstly we will make some basic preparation for attractors on time-dependent space, see [7,8] for more details, and then, we will recall some properties of the nonlinear damping g and the nonlinearity f. Finally, we state the results about the well-posedness of problem (1.1).

For convenience, C denotes a general positive constant which changes in various situation, it can be replaced by $C_{1}, C_{2}, \dots$ . The norm in $L^{2} (Ω)$ is denoted by $‖ \cdot ‖$ and $⟨, ⟩$ denotes the inner product in $L^{2} (Ω)$ . Denote the time-dependent space $H_{t} = H_{0}^{2} (Ω) \times L^{2} (Ω)$ with norm $‖ (u, u_{t}) ‖_{H_{t}}^{2} = ‖ Δ u ‖^{2} + ε (t) ‖ u_{t} ‖^{2} + λ ‖ u ‖^{2}$ .

2.1. Notations and abstract results

Let ${X_{t}}_{t \in R}$ be a family of normed spaces, we denote the R-ball of $X_{t}$ by $\begin{matrix} B_{t} (R) = {x \in X_{t} : ‖ x ‖_{X_{t}} ⩽ R} . \end{matrix}$ The Hausdorff semidistance of two nonempty sets $A, B \subset X_{t}$ is defined as $\begin{matrix} δ_{t} (A, B) = sup_{x \in A} inf_{y \in B} ‖ x - y ‖_{X_{t}} . \end{matrix}$ The ϵ-neighborhood of a set $C \subset X_{t}$ is denoted by $\begin{matrix} O_{t}^{ϵ} (C) = ⋃_{x \in C} {y \in X_{t} : ‖ x - y ‖_{X_{t}} < ϵ} = ⋃_{x \in C} {x + B_{t} (ϵ)} . \end{matrix}$

Definition 2.1.
Let ${X_{t}}_{t \in R}$ be a family of normed spaces. A process is two-parameter family of mapping ${U (t, τ) : X_{τ} \to X_{t}, t ⩾ τ, τ \in R}$ with properties:
$U (τ, τ)$ is the identity operator on $X_{τ}$ ;

$U (t, s) U (s, τ) = U (t, τ)$ , $\forall t ⩾ s ⩾ τ$ .

Definition 2.2.
A family ${C_{t}}_{t \in R}$ of bounded sets $C_{t} \subset X_{t}$ is called uniformly bounded if there exists $R > 0$ such that $\begin{matrix} C_{t} \subset B_{t} (R), \forall t \in R . \end{matrix}$
Definition 2.3.
A family ${B_{t}}_{t \in R}$ is called pullback absorbing if it is uniformly bounded and for every $R > 0$ , there exists a constant $t_{0} = t_{0} (t, R) ⩽ t$ such that $\begin{matrix} τ ⩽ t_{0} \Rightarrow U (t, τ) B_{τ} (R) \subset B_{t} . \end{matrix}$ Moreover, we say the process is dissipative when it admits a pullback absorbing family.
Definition 2.4.
A time-dependent absorbing set for the process $U (t, τ)$ is a uniformly bounded family ${B_{t}}_{t \in R}$ satisfying the following property: for every $R ⩾ 0$ there exists a $t_{0} = t_{0} (R)$ such that $\begin{matrix} τ ⩽ t - t_{0} \Rightarrow U (t, τ) B_{τ} (R) \subset B_{t} . \end{matrix}$
Definition 2.5.
The time-dependent global attractor for $U (t, τ)$ is the smallest family ${A_{t}}_{t \in R}$ such that
$A_{t}$ is compact in $X_{t}$ , $\forall t \in R$ ;

The family ${A_{t}}_{t \in R}$ is pullback attracting, i.e. it is uniformly bounded and the limit $\begin{matrix} lim_{τ \to - \infty} δ_{t} (U (t, τ) C_{τ}, A_{t}) = 0 \end{matrix}$
holds for every uniformly bounded family ${C_{t}}_{t \in R}$ and every fixed $t \in R$ .
Definition 2.6.
The time-dependent global attractor ${A_{t}}_{t \in R}$ for $U (t, τ)$ is invariant if $\begin{matrix} U (t, τ) A_{τ} = A_{t}, \forall t ⩾ τ . \end{matrix}$
Definition 2.7.
A process $U (\cdot, \cdot)$ in a family of normed spaces ${X_{t}}_{t \in R}$ is called pullback asymptotically compact if and only if for any fixed $t \in R$ , bounded sequence ${x_{n}}_{n = 1}^{\infty}$ such that $x_{n} \in X_{τ_{n}}$ and any ${τ_{n}}_{n = 1}^{\infty} \subset R^{- t}$ with $τ_{n} \to - \infty$ as $n \to \infty$ , the sequence ${U (t, τ_{n}) x_{n}}_{n = 1}^{\infty}$ has a convergent subsequence, where $R^{- t} = {τ : τ \in R, τ ⩽ t}$ .
Theorem 2.8 ([15]).

Let $U (\cdot, \cdot)$ be a process in a family of Banach spaces ${X_{t}}_{t \in R}$ , then $U (\cdot, \cdot)$ has a time-dependent global attractor ${A_{t}}_{t \in R}$ satisfying $\begin{matrix} A_{t} = ⋂_{s ⩽ t} \overline{⋃_{τ ⩽ s} U (t, τ) B_{τ}}, \end{matrix}$ if and only if

$U (\cdot, \cdot)$ has a pullback absorbing family ${B_{t}}_{t \in R}$ ;

$U (\cdot, \cdot)$ is pullback asymptotically compact.

We now introduce the technical method via contractive functions to verify the pullback asymptotic compactness on time-depend spaces. This technique to prove compactness of autonomous system was initiated by Khanmamedov [10] and Chueshov, Lasiecka [5]. In the following we apply this method to time-depend space.

Definition 2.9.
Let ${X_{t}}_{t \in R}$ be a family Banach spaces and ${B_{t}}_{t \in R}$ be a family subsets of ${X_{t}}_{t \in R}$ , uniformly bounded. We say a function $ϕ_{τ}^{t} (\cdot, \cdot)$ , defined on $X_{t} \times X_{t}$ , a contractive function on $B_{τ} \times B_{τ}$ if for any fixed $t \in R$ and any sequence ${x_{n}}_{n = 1}^{\infty} \subset B_{τ}$ , there exists a subsequence ${x_{n_{k}}}_{k = 1}^{\infty} \subset {x_{n}}_{n = 1}^{\infty}$ such that $\begin{matrix} lim_{k \to \infty} lim_{l \to \infty} ϕ_{τ}^{t} (x_{n_{k}}, x_{n_{l}}) = 0, \end{matrix}$ where $τ ⩽ t$ . We denote the set of all contractive functions on $B_{τ} \times B_{τ}$ by $C (B_{τ})$ .
Theorem 2.10 ([15]).

Let $U (\cdot, \cdot)$ be a process on ${X_{t}}_{t \in R}$ and has a pullback absorbing family ${B_{t}}_{t \in R}$ . And assume that for any $ϵ > 0$ there exists $T (ϵ) ⩽ t$ , $ϕ_{T}^{t} \in C (B_{T})$ such that $\begin{matrix} ‖ U (t, T) x - U (t, T) y ‖ ⩽ ϵ + ϕ_{T}^{t} (x, y), \end{matrix}$ for any $x, y \in B_{T}$ and any fixed $t \in R$ . Then $U (\cdot, \cdot)$ is pullback asymptotically compact.

2.2. Some properties for nonlinear function f and g

From the assumptions on g and applying Young’s inequality, we have $\begin{matrix} (2.1) & {| g (s) |}^{\frac{q + 1}{q}} ⩽ γ + C_{γ} g (s) s, \end{matrix}$ where $γ > 0$ is arbitrary and $C_{γ} > 0$ depends on γ. Moreover, $\begin{matrix} (2.2) & | g (s) | ⩽ C + C {(g (s) s)}^{\frac{q}{q + 1}} . \end{matrix}$

Lemma 2.11 ([10]).

Let g satisfy ( 1.7 ). Then for any $ϑ > 0$ , there exists $C_{ϑ} > 0$ depending on ϑ such that $\begin{matrix} (2.3) & | u - v |^{2} ⩽ ϑ + C_{ϑ} (g (u) - g (v)) (u - v) for all u, v \in R . \end{matrix}$

Set $\begin{matrix} F (u) = \int_{0}^{u (x)} f (y) d y . \end{matrix}$ We obtain the following inequality from (1.6), $\begin{array}{l} (2.4) & \int_{Ω} F (u) d x ⩾ - \frac{λ_{2}}{2} ‖ u ‖^{2} - C, \\ (2.5) & ⟨ f (u), u ⟩ ⩾ \int_{Ω} F (u) d x - \frac{λ_{2}}{2} ‖ u ‖^{2} - C, for some λ_{2} ⩽ λ_{1} . \end{array}$

2.3. Well-posedness

In the following we state the well-posedness of problem (1.1). As mentioned in [3,11], we can obtain the existence of solution by using Faedo–Galerkin method. Where $ε (t)$ depends on time t, which will bring some difficulties in energy estimates, see Theorem 3.1 below for more details.

Theorem 2.12.
Under the assumptions ( 1.1 )–( 1.8 ), for any initial data $(u_{0} (τ), u_{1} (τ)) \in H_{τ}$ , on any interval $[τ, t]$ with $t > τ$ , then there exists a unique weak solution $\begin{matrix} u \in C ([τ, t], H_{0}^{2} (Ω)), u_{t} \in C ([τ, t], L^{2} (Ω)), \end{matrix}$ which continuously depend on the initial data. In other words, problem ( 1.1 )–( 1.2 ) generate a strongly continuous process $U (t, τ) : H_{τ} \to H_{t}$ , $t ⩾ τ \in R$ , such that $\begin{matrix} U (t, τ) (u_{0} (τ), u_{1} (τ)) = (u (t), u_{t} (t)) . \end{matrix}$
Proof.
The existence of weak solution of problem (1.1)–(1.2) is classical and it can be obtained by Faedo–Galerkin method. We won’t repeat it here.

Given two different initial datum $z_{1} (τ), z_{2} (τ) \in H_{τ}$ such that $‖ z_{i} ‖_{H_{τ}} ⩽ R$ , $i = 1, 2$ . By the below Theorem 3.1 we know that $\begin{matrix} (2.6) & {‖ U (t, τ) z_{i} (τ) ‖}_{H_{t}} ⩽ C . \end{matrix}$ We denote $\begin{array}{l} (u_{i} (t), \partial_{t} u_{i} (t)) = U (t, τ) z_{i} (τ), \bar{z} (t) = U (t, τ) z_{1} (τ) - U (t, τ) z_{2} (τ), \\ \bar{u} (t) = u_{1} (t) - u_{2} (t), \end{array}$ so we obtain $\bar{u}$ satisfies $\begin{matrix} \{\begin{matrix} ε {\bar{u}}_{t t} + g (u_{1 t}) - g (u_{2 t}) + Δ^{2} (\bar{u}) + λ \bar{u} + f (u_{1}) - f (u_{2}) = 0, & t > τ, \\ \bar{u} |_{\partial Ω} = 0, \\ (\bar{u} (x, τ), {\bar{u}}_{t} (x, τ)) = z_{1} (τ) - z_{2} (τ) . \end{matrix} \end{matrix}$ Multiplying the above equality by $2 {\bar{u}}_{t}$ and integrating over Ω, we have $\begin{matrix} (2.7) & \frac{d}{d t} ‖ \bar{z} ‖_{H_{t}}^{2} + 2 \int_{Ω} (g (u_{1 t} (t)) - g (u_{1 t} (t))) {\bar{u}}_{t} d x - ε^{'} ‖ {\bar{u}}_{t} ‖^{2} = - 2 ⟨ f (u_{1}) - f (u_{2}), {\bar{u}}_{t} ⟩ . \end{matrix}$ From (1.7) we obtain $\begin{matrix} 2 \int_{Ω} (g (u_{1 t}) - g (u_{2 t})) {\bar{u}}_{t} d x ⩾ 0 . \end{matrix}$ According to (1.5) we have $\begin{matrix} - 2 ⟨ f (u_{1}) - f (u_{2}), {\bar{u}}_{t} ⟩ ⩽ C ⟨ (1 + u_{1}^{p} + u_{2}^{p}) \bar{u}, u_{t} ⟩ . \end{matrix}$ From the embedding inequalities (since $p ⩽ \frac{4}{n - 4}$ ), we have $\begin{array}{l} ‖ u ‖_{L^{2 p + 2}} ⩽ C ‖ u ‖_{H_{0}^{2}}, \end{array}$ where C depends on Ω. Combining with the above inequalities, (2.6) and general Hölder’s inequality we have $\begin{array}{l} ⟨ | u_{i} |^{p} \bar{u}, {\bar{u}}_{t} ⟩ & ⩽ C ‖ u_{i} ‖_{L^{2 p + 2}}^{p} ‖ \bar{u} ‖_{H_{0}^{2}} ‖ {\bar{u}}_{t} ‖ \\ ⩽ C ‖ \bar{u} ‖_{H_{0}^{2}} ‖ {\bar{u}}_{t} ‖ \\ (2.8) & ⩽ C ‖ Δ \bar{u} ‖^{2} + C ‖ {\bar{u}}_{t} ‖^{2}, \end{array}$ it follows from (1.4), (2.7), Young’s inequality and $ε^{'} (t) ⩽ 0$ that $\begin{array}{l} \frac{d}{d t} {‖ \bar{z} (t) ‖}_{H_{t}}^{2} & ⩽ C ⟨ (1 + u_{1}^{p} + u_{2}^{p}) \bar{u}, u_{t} ⟩ \\ ⩽ C ⟨ | u_{1} |^{p} \bar{u}, {\bar{u}}_{t} ⟩ + C ⟨ | u_{2} |^{p} \bar{u}, {\bar{u}}_{t} ⟩ + C ‖ \bar{u} ‖^{2} + C ‖ u_{t} ‖^{2} \\ ⩽ C ‖ Δ \bar{u} ‖^{2} + C ‖ {\bar{u}}_{t} ‖^{2} \\ ⩽ C \frac{1}{ε (t)} (L ‖ Δ \bar{u} ‖^{2} + ε (t) ‖ {\bar{u}}_{t} ‖^{2}) \\ (2.9) & ⩽ C \frac{1}{ε (t)} {‖ \bar{z} (t) ‖}_{H_{t}}^{2} . \end{array}$ Applying the Gronwall lemma on $[τ, t]$ , we obtain $\begin{matrix} {‖ \bar{z} (t) ‖}_{H_{t}}^{2} ⩽ e^{C \int_{τ}^{t} \frac{1}{ε (s)} d s} {‖ \bar{z} (τ) ‖}_{H_{τ}}^{2}, \end{matrix}$ the proof is completed. □

3. Absorbing sets

In this section, we will prove the dissipativity of equation (1.1), the approach is mainly inspired by [13,15].

Theorem 3.1.
Under assumptions ( 1.1 )–( 1.8 ), for any initial data $(u_{0} (τ), u_{1} (τ)) \in B_{τ} (R) \subset H_{τ}$ , there exists $R_{0} > 0$ such that the family ${B_{t} (R_{0})}_{t \in R}$ is a time-dependent absorbing set for the process $U (t, τ)$ corresponding to ( 1.1 ).
Lemma 3.2.
Assume that $t ⩾ τ$ and initial datum $z \in H_{τ}$ , there exist positive constants $C_{1}$ , δ and a positive increasing function Q such that $\begin{matrix} (3.1) & {‖ U (t, τ) z ‖}_{H_{t}} ⩽ Q (‖ z ‖_{H_{τ}}) e^{- δ (t - τ)} + C_{1} . \end{matrix}$
Proof.
Denote $\begin{matrix} ε (t) = \frac{1}{2} {‖ U (t, τ) z ‖}_{H_{t}}^{2} \end{matrix}$ and $\begin{matrix} E (t) = ε (t) + \int_{Ω} (F (u) - h u) d x . \end{matrix}$ Multiplying (1.1) by $u_{t}$ and integrating over Ω, we obtain $\begin{matrix} (3.2) & \frac{d}{d t} E (t) + \int_{Ω} g (u_{t}) u_{t} d x - \int_{Ω} \frac{ε^{'}}{2} | u_{t} |^{2} d x = 0 . \end{matrix}$ Since $ε^{'} (t) ⩽ 0$ and (1.7), we have $\begin{matrix} (3.3) & E (t) is decreasing, E (t) ⩽ E (τ), \forall t ⩾ τ . \end{matrix}$ From (1.6) and Sobolev’s embedding, there exist some proper positive constant $c_{0}$ , $C_{0}$ and the function $C (s)$ such that $\begin{matrix} (3.4) & c_{0} ε (t) - C_{0} ⩽ E (t) ⩽ C (ε (t)) . \end{matrix}$ Meanwhile, combining with (2.2), Hölder’s inequality and Young’s inequality, we have $\begin{array}{l} | \int_{Ω} g (u_{t}) u d x | & ⩽ C \int_{Ω} | u | d x + C \int_{Ω} {(g (u_{t}) u_{t})}^{\frac{q}{q + 1}} | u | d x \\ ⩽ C \int_{Ω} | u | d x + C {(\int_{Ω} g (u_{t}) u_{t} d x)}^{\frac{q}{q + 1}} {(\int_{Ω} | u |^{q + 1} d x)}^{\frac{1}{q + 1}} \\ ⩽ C \int_{Ω} | u | d x + C {(\int_{Ω} g (u_{t}) u_{t} d x)}^{\frac{q}{q + 1}} ‖ Δ u ‖ \\ ⩽ C \int_{Ω} | u | d x + C (1 + \int_{Ω} g (u_{t}) u_{t} d x) ‖ Δ u ‖ \\ (3.5) & ⩽ M_{g, Ω} + η ‖ Δ u ‖^{2} + C (ε (τ)) \int_{Ω} g (u_{t}) u_{t} d x, \end{array}$ where $η > 0$ is small enough and $M_{g, Ω}$ is a constant different in various situations.

Denote $\begin{matrix} V (t) = E (t) + δ ε (t) ⟨ u_{t}, u ⟩ . \end{matrix}$ Differentiating $V (t)$ and combining with (3.2), we obtain $\begin{array}{l} \frac{d V (t)}{d t} = & \frac{d E (t)}{d t} + δ ε (t) ⟨ u_{t t}, u ⟩ + δ ε^{'} (t) ⟨ u_{t}, u ⟩ + δ ε (t) ⟨ u_{t}, u_{t} ⟩ \\ = & - \int_{Ω} g (u_{t}) u_{t} d x + \frac{1}{2} \int_{Ω} ε^{'} u_{t}^{2} d x + δ ε^{'} (t) ⟨ u_{t}, u ⟩ \\ (3.6) & + δ ε (t) ⟨ u_{t}, u_{t} ⟩ - δ ⟨ g (u_{t}) + Δ^{2} u + λ u + f (u) - h (u), u ⟩ . \end{array}$ So, we have $\begin{array}{rcl} \frac{d V (t)}{d t} + δ V (t) & = & - \int_{Ω} g (u_{t}) u_{t} d x + \frac{1}{2} \int_{Ω} ε^{'} u_{t}^{2} d x + δ ε^{'} (t) ⟨ u_{t}, u ⟩ + \frac{3}{2} δ ε ⟨ u_{t}, u_{t} ⟩ \\ - δ ⟨ g (u_{t}), u ⟩ - \frac{δ}{2} ‖ Δ u ‖^{2} - \frac{λ δ}{2} ‖ u ‖^{2} - δ ⟨ f (u), u ⟩ + δ \int_{Ω} F (u) d x \\ (3.7) & + δ^{2} ε ⟨ u_{t}, u ⟩ . \end{array}$ Next, we estimate the right term. According to (1.6), (2.5) and (3.5),we obtain $\begin{array}{rcl} \frac{d V (t)}{d t} + δ V (t) & ⩽ & - \int_{Ω} g (u_{t}) u_{t} d x + \frac{1}{2} \int_{Ω} ε^{'} u_{t}^{2} d x + δ ε^{'} (t) ⟨ u_{t}, u ⟩ + \frac{3}{2} δ ε ⟨ u_{t}, u_{t} ⟩ \\ + δ^{2} ε ⟨ u_{t}, u ⟩ + δ C (ε (τ)) \int_{Ω} g (u_{t}) u_{t} d x \\ (3.8) & - \frac{δ}{2} (λ_{1} - λ_{2} - 2 η) ‖ u ‖^{2} + δ M_{g, Ω} + C . \end{array}$ Combining the property of $ε (t)$ , Hölder inequality with Young’s inequality, we get $\begin{array}{l} (3.9) & | ε^{'} (t) ⟨ u_{t}, u ⟩ | ⩽ L ‖ u_{t} ‖ ‖ u ‖ ⩽ \frac{L^{2} ‖ u_{t} ‖^{2}}{4 η} + η ‖ u ‖^{2}, \\ (3.10) & | ε (t) ⟨ u_{t}, u ⟩ | ⩽ L ‖ u_{t} ‖ ‖ u ‖ ⩽ \frac{L^{2} ‖ u_{t} ‖^{2}}{4 η} + η ‖ u ‖^{2} . \end{array}$ From (3.8), (3.9) and (3.10), we obtain $\begin{array}{rcl} \frac{d V (t)}{d t} + δ V (t) & ⩽ & (δ C (ε (τ)) - 1) \int_{Ω} g (u_{t}) u_{t} d x + \frac{1}{2} ε^{'} (t) \int_{Ω} u_{t}^{2} d x \\ + δ (\frac{3}{2} ε + \frac{L^{2}}{4 η} + \frac{δ L^{2}}{4 η}) \int_{Ω} u_{t}^{2} d x \\ (3.11) & - \frac{δ}{2} (λ_{1} - λ_{2} - 4 η - 2 δ η) ‖ u ‖^{2} + δ M_{g, Ω} + C . \end{array}$ In order to apply the Gronwall lemma, we need to estimate the right term of (3.11). By using of (1.7), there exist $m > 0$ and $R_{g}$ such that $\begin{matrix} g^{'} ⩾ m, when | s | > R_{g}, \end{matrix}$ therefore, $\begin{matrix} (3.12) & \int_{Ω} g (u_{t}) u_{t} d x ⩾ m \int_{Ω {| u_{t} | ⩾ R_{g}}} | u_{t} |^{2} d x . \end{matrix}$ Then for $δ_{1}$ small enough, we have $\begin{array}{l} \int_{Ω} g (u_{t}) u_{t} d x - δ_{1} ‖ u_{t} ‖^{2} \\ ⩾ m \int_{Ω {| u_{t} | ⩾ R_{g}}} | u_{t} |^{2} d x - δ_{1} \int_{Ω {| u_{t} | ⩾ R_{g}}} | u_{t} |^{2} d x - δ_{1} \int_{Ω {| u_{t} | ⩽ R_{g}}} | u_{t} |^{2} d x \\ (3.13) & ⩾ (m - δ_{1}) \int_{Ω {| u_{t} | ⩾ R_{g}}} | u_{t} |^{2} d x + δ_{1} \int_{Ω {| u_{t} | ⩽ R_{g}}} | u_{t} |^{2} d x - 2 δ_{1} R_{g}^{2} | Ω | . \end{array}$ Let $δ = η^{2}$ , η be small enough, we can conclude $\begin{matrix} \int_{Ω} g (u_{t}) u_{t} d x - δ (\frac{3}{2} ε + \frac{L^{2}}{4 η} + \frac{δ L^{2}}{4 η}) \int_{Ω} u_{t}^{2} d x ⩾ 0, \end{matrix}$ thus, $\begin{matrix} (3.14) & \frac{d V (t)}{d t} + δ V (t) ⩽ δ M_{g, Ω} . \end{matrix}$ Using Gronwall lemma on $[τ, t]$ , we have $\begin{matrix} (3.15) & V (t) ⩽ V (τ) e^{- δ (t - τ)} + C . \end{matrix}$ Since $\begin{matrix} (3.16) & δ ε | ⟨ u_{t}, u ⟩ | ⩽ \frac{ε}{4} ‖ u_{t} ‖^{2} + L δ^{2} ‖ u ‖^{2}, \end{matrix}$ by Young’s inequality and Hölder inequality, and combining (3.4) with (3.16), we obtain $\begin{matrix} (3.17) & c_{0} ε (t) - C_{0} ⩽ V (t) ⩽ C (ε (t)) . \end{matrix}$ Therefore, there exist a constant $C_{1} > 0$ and a positive increasing function Q such that $\begin{matrix} (3.18) & {‖ U (t, τ) z ‖}_{H_{t}} ⩽ Q (‖ z ‖_{H_{τ}}) e^{- δ (t - τ)} + C_{1} . \end{matrix}$ □
Proof of Theorem 3.1.
For any $z \in B_{τ} (R)$ , let $t_{0} = max {0, δ^{- 1} ln \frac{Q (R)}{1 + C_{1}}}$ , then we have $\begin{matrix} {‖ U (t, τ) z ‖}_{H_{t}} ⩽ Q (R) e^{- δ (t - τ)} + C_{1} ⩽ 1 + 2 C_{1} ≜ R_{0}, \end{matrix}$ when $t - τ ⩾ t_{0}$ . Therefore, ${B_{t} (R_{0})}_{t \in R}$ is pullback absorbing and the process is dissipative. □

4. Asymptotically compact

In this section, we will show that the process $U (\cdot, \cdot)$ corresponding to the equation (1.1) is pullback asymptotically compact. We complete the proof by two steps, firstly we make some priori estimates, and then we will verify the asymptotic compactness of the process by using contractive function. Let $(u_{i} (t), u_{i t} (t))$ be the corresponding solution of (1.1) with initial data $(u_{0}^{i} (τ), v_{0}^{i} (τ)) \in {B_{τ}}_{τ \in R}$ . Here are some notations: $\begin{matrix} w (t) = u_{1} (t) - u_{2} (t), g_{i} (t) = g (u_{i t} (t)), f_{i} (t) = f (u_{i} (t)) i = 1, 2 . \end{matrix}$ Then $w (t)$ satisfies $\begin{matrix} (4.1) & \{\begin{matrix} ε w_{t t} + g_{1} (t) - g_{2} (t) + Δ^{2} w + λ w + f_{1} (t) - f_{2} (t) = 0, t > T; \\ w |_{\partial Ω} = \frac{\partial}{\partial v} w |_{\partial Ω} = 0; \\ w (x, T) = u_{0}^{1} (T) - u_{0}^{2} (T) w_{t} (x, T) = v_{0}^{1} (T) - v_{0}^{2} (T) . \end{matrix} \end{matrix}$ Denote $\begin{matrix} ε_{w} (t) = \frac{1}{2} \int_{Ω} [ε (t) {| w_{t} (t) |}^{2} + {| Δ w (t) |}^{2} + λ | w |^{2}] d x . \end{matrix}$

Lemma 4.1.

Let ${B_{τ}}_{τ \in R}$ be the time-dependent absorbing family given by 3.1 . Then there exists $C_{δ} > 0$ such that $\begin{array}{l} (t - T) ε_{w} (t) \\ ⩽ - \int_{T}^{t} \int_{s}^{t} \int_{Ω} (f_{1} (ξ) - f_{2} (ξ)) w_{t} (ξ) d x d ξ d s \\ + \frac{1}{2} L δ mes (Ω) (t - T) + \frac{1}{2} L C_{δ} ε_{w} (T) - \frac{1}{2} L C_{δ} \int_{T}^{t} \int_{Ω} (f_{1} (ξ) - f_{2} (ξ)) w_{t} (ξ) d x d ξ \\ - \frac{1}{2} \int_{Ω} ε (t) w_{t} w d x + \frac{1}{2} \int_{T}^{t} \int_{Ω} L | w_{t} w | d x d ξ - \frac{1}{2} \int_{T}^{t} \int_{Ω} (g_{1} (ξ) - g_{2} (ξ)) w d x d ξ \\ (4.2) & + \frac{1}{2} \int_{Ω} ε (T) w_{t} (T) w (T) d x - \frac{1}{2} \int_{T}^{t} \int_{Ω} (f_{1} (ξ) - f_{2} (ξ)) w (ξ) d x d ξ . \end{array}$

Proof.

Multiplying (4.1) by $w_{t}$ , and integrating over $[s, t] \times Ω$ , we obtain $\begin{array}{l} \int_{s}^{t} \int_{Ω} [(g_{1} (ξ) - g_{2} (ξ)) w_{t} (ξ) - \frac{1}{2} ε^{'} (ξ) {| w_{t} (ξ) |}^{2} + (f_{1} (ξ) - f_{2} (ξ)) w_{t} (ξ)] d x d ξ \\ (4.3) & + ε_{w} (t) - ε_{w} (s) = 0, \end{array}$ where $T ⩽ s ⩽ t$ , then we have $\begin{array}{l} \int_{s}^{t} \int_{Ω} [(g_{1} (ξ) - g_{2} (ξ)) w_{t} (ξ) - \frac{1}{2} ε^{'} (ξ) {| w_{t} (ξ) |}^{2}] d x d ξ \\ (4.4) & ⩽ ε_{w} (s) - \int_{s}^{t} \int_{Ω} (f_{1} (ξ) - f_{2} (ξ)) w_{t} (ξ) d x d ξ . \end{array}$ Note that $ε (t)$ is decreasing and combine with (2.3), for any $δ > 0$ , there exists $C_{δ} > 0$ such that $\begin{matrix} (4.5) & ε (ξ) | w_{t} |^{2} ⩽ L | w_{t} |^{2} ⩽ L δ + L C_{δ} ((g_{1} (ξ) - g_{2} (ξ)) w_{t} (ξ) - \frac{1}{2} ε^{'} (ξ) | w_{t} |^{2}) . \end{matrix}$ By (4.4) and (4.5), we can obtain $\begin{array}{l} \int_{s}^{t} \int_{Ω} ε (ξ) {| w_{t} (ξ) |}^{2} d x d ξ ⩽ & L δ mes (Ω) (t - s) + L C_{δ} ε_{w} (s) \\ (4.6) & - L C_{δ} \int_{s}^{t} \int_{Ω} (f_{1} (ξ) - f_{2} (ξ)) w_{t} (ξ) d x d ξ . \end{array}$ Multiplying (4.1) by $w (t)$ and integrating over $[T, t] \times Ω$ , we get $\begin{array}{l} \int_{T}^{t} \int_{Ω} {| Δ w (s) |}^{2} d x d s + \int_{Ω} ε (t) w_{t} w d x + \int_{T}^{t} \int_{Ω} λ w^{2} d x d s \\ = \int_{T}^{t} \int_{Ω} ε (ξ) {| w_{t} (ξ) |}^{2} d x d ξ + \int_{T}^{t} \int_{Ω} ε^{'} (ξ) w_{t} w d x d ξ \\ - \int_{T}^{t} \int_{Ω} (g_{1} (ξ) - g_{2} (ξ)) w d x d ξ \\ (4.7) & + \int_{Ω} ε (T) w_{t} (T) w (T) d x - \int_{T}^{t} \int_{Ω} (f_{1} (ξ) - f_{2} (ξ)) w (ξ) d x d ξ . \end{array}$ Combining (4.6) with (4.7), we immediately know that $\begin{array}{l} 2 \int_{T}^{t} ε_{w} (ξ) d ξ ⩽ & L δ mes (Ω) (t - T) + L C_{δ} ε_{w} (T) \\ - L C_{δ} \int_{T}^{t} \int_{Ω} (f_{1} (ξ) - f_{2} (ξ)) w_{t} (ξ) d x d ξ - \int_{Ω} ε (t) w_{t} w d x \\ + \int_{T}^{t} \int_{Ω} L | w_{t} w | d x d ξ - \int_{T}^{t} \int_{Ω} (g_{1} (ξ) - g_{2} (ξ)) w (ξ) d x d ξ \\ (4.8) & + \int_{Ω} ε (T) w_{t} (T) w (T) d x - \int_{T}^{t} \int_{Ω} (f_{1} (ξ) - f_{2} (ξ)) w (ξ) d x d ξ . \end{array}$ Integrating (4.3) over $[T, t]$ with respect to s, we have $\begin{array}{l} (t - T) ε_{w} (t) + \int_{T}^{t} \int_{s}^{t} \int_{Ω} [(g_{1} (ξ) - g_{2} (ξ)) w_{t} (ξ) - \frac{1}{2} ε^{'} (ξ) {| w_{t} (ξ) |}^{2}] d x d ξ d s \\ (4.9) & = - \int_{T}^{t} \int_{s}^{t} \int_{Ω} (f_{1} (ξ) - f_{2} (ξ)) w_{t} (ξ) d x d ξ d s + \int_{T}^{t} ε_{w} (s) d s . \end{array}$ Note that $ε (t)$ is decreasing and apply (1.7), we can get $\begin{matrix} (4.10) & (g_{1} (ξ) - g_{2} (ξ)) w_{t} (ξ) - \frac{1}{2} ε^{'} (ξ) | w_{t} |^{2} ⩾ 0 . \end{matrix}$ It is obvious that the inequality (4.2) follows frow (4.8), (4.9) and (4.10). □

Theorem 4.2.

Under the assumptions ( 1.2 )–( 1.8 ) for any fixed $t \in R$ , bounded sequence ${x_{n}}_{n = 1}^{\infty} \subset X_{τ_{n}}$ and any ${τ_{n}}_{n = 1}^{\infty} \subset R^{- t}$ with $τ_{n} \to - \infty$ as $n \to \infty$ , the sequence ${U (t, τ_{n}) x_{n}}_{n = 1}^{\infty}$ has a convergent subsequence.

Proof.

By using the dissipation integral $\begin{matrix} (4.11) & \int_{τ}^{\infty} (g (\partial_{t} u (t)), \partial_{t} u (t)) d t ⩽ C, \end{matrix}$ where the constant C is independent of τ and the initial data from the absorbing ball $B_{τ} (R)$ , we can obtain $\begin{matrix} (4.12) & \int_{T}^{t} \int_{Ω} g (u_{i t}) u_{i t} d x d ξ ⩽ C^{'}, \end{matrix}$ where the constant $C^{'}$ is independent of T. By using of the property of g and the above inequality (4.12), we get $\begin{matrix} (4.13) & \int_{T}^{t} \int_{Ω} {| g (u_{i t}) |}^{\frac{q + 1}{q}} d x d ξ ⩽ C_{γ} + γ (t - T) mes (Ω), \end{matrix}$ where $C_{γ}$ is a constant which depends on $C^{'}$ and γ. Therefore, using Hölder inequality and (4.13) we have $\begin{array}{l} | \int_{T}^{t} \int_{Ω} g_{i} w d x d ξ | & ⩽ {(\int_{T}^{t} \int_{Ω} {| g (u_{i t}) |}^{\frac{q + 1}{q}} d x d ξ)}^{\frac{q}{q + 1}} {(\int_{T}^{t} \int_{Ω} | w |^{q + 1} d x d ξ)}^{\frac{1}{q + 1}} \\ ⩽ (C_{γ} + {(γ (t - T) mes (Ω))}^{\frac{q}{q + 1}}) {(\int_{T}^{t} \int_{Ω} | w |^{q + 1} d x d ξ)}^{\frac{1}{q + 1}} \\ (4.14) & ⩽ C_{γ} + C_{ρ} {(γ mes (Ω))}^{\frac{q}{q + 1}} (t - T), \end{array}$ where $C_{ρ}$ is a constant which depends on the size of the bounded absorbing set, but is independent of T. Moreover, we get $\begin{matrix} (4.15) & | \int_{T}^{t} \int_{Ω} (g_{1} (ξ) - g_{2} (ξ)) w d x d ξ | ⩽ C_{γ} + C_{ρ} {(γ mes (Ω))}^{\frac{q}{q + 1}} (t - T) . \end{matrix}$ From (4.2) and (4.15), we can obtain $\begin{array}{l} (t - T) ε_{w} (t) ⩽ & \frac{1}{2} L δ mes (Ω) (t - T) + \frac{1}{2} L C_{δ} ε_{w} (T) + \frac{1}{2} \int_{Ω} ε (T) w_{t} (T) w (T) d x \\ - \int_{T}^{t} \int_{s}^{t} \int_{Ω} (f_{1} (ξ) - f_{2} (ξ)) w_{t} (ξ) d x d ξ d s - \frac{1}{2} \int_{Ω} ε (t) w_{t} w d x \\ + \frac{1}{2} \int_{T}^{t} \int_{Ω} L | w_{t} w | d x d ξ - \frac{1}{2} L C_{δ} \int_{T}^{t} \int_{Ω} (f_{1} (ξ) - f_{2} (ξ)) w_{t} (ξ) d x d ξ \\ + C_{γ} + C_{ρ} {(γ mes (Ω))}^{\frac{q}{q + 1}} (t - T) \\ (4.16) & - \frac{1}{2} \int_{T}^{t} \int_{Ω} (f_{1} (ξ) - f_{2} (ξ)) w (ξ) d x d ξ . \end{array}$ In the above inequality (4.16), we set $\begin{array}{l} ϕ_{T}^{t} ((u_{0}^{1} (T), v_{0}^{1} (T)), (u_{0}^{2} (T), v_{0}^{2} (T))) \\ ≜ - \frac{1}{t - T} \int_{T}^{t} \int_{s}^{t} \int_{Ω} (f_{1} (ξ) - f_{2} (ξ)) w_{t} (ξ) d x d ξ d s \\ - \frac{1}{2 (t - T)} \int_{Ω} ε (t) w_{t} w d x + \frac{1}{2 (t - T)} \int_{T}^{t} \int_{Ω} L | w_{t} w | d x d ξ \\ - \frac{L C_{δ}}{2 (t - T)} \int_{T}^{t} \int_{Ω} (f_{1} (ξ) - f_{2} (ξ)) w_{t} (ξ) d x d ξ \\ (4.17) & - \frac{1}{2 (t - T)} \int_{s}^{t} \int_{Ω} (f_{1} (ξ) - f_{2} (ξ)) w (ξ) d x d ξ \end{array}$ and $\begin{array}{l} C_{M} ≜ & \frac{1}{2} L δ mes (Ω) (t - T) + \frac{1}{2} L C_{δ} ε_{w} (T) \\ (4.18) & + \frac{1}{2} \int_{Ω} ε (T) w_{t} (T) w (T) d x + C_{γ} + C_{ρ} {(γ mes (Ω))}^{\frac{q}{q + 1}} (t - T) . \end{array}$ So, we obtain $\begin{matrix} (4.19) & ε_{w} (t) ⩽ \frac{C_{M}}{t - T} + ϕ_{T}^{t} ((u_{0}^{1} (T), v_{0}^{1} (T)), (u_{0}^{2} (T), v_{0}^{2} (T))) . \end{matrix}$ For any fixed $ϵ > 0$ , we first choose some proper $δ > 0$ and $γ > 0$ , such that $\begin{matrix} \frac{1}{2} L δ mes (Ω) + C_{ρ} {(γ mes (Ω))}^{\frac{q}{q + 1}} ⩽ \frac{ϵ}{4}, \end{matrix}$ for some fixed t, let $T < t$ and $t - T$ large enough such that $\begin{matrix} \frac{C_{M}}{t - T} < \frac{ϵ}{2} . \end{matrix}$ Due to Theorem 2.10, we only need to verify that $ϕ_{T}^{t} \in C (B_{t})$ for each fixed T.

Let $(u_{n}, u_{n t})$ be the solution corresponding to initial data $(u_{0}^{n}, v_{0}^{n})$ . From the dissipative of problem (1.1), we obtain that $‖ Δ u_{n} ‖^{2} + ε (ξ) ‖ u_{n t} ‖^{2} + λ ‖ u_{n} ‖^{2}$ is bounded for fixed T, $ξ \in [T, t]$ . According to Alaoglu theorem, without loss of generality (at most by passing to subsequence) we deduce that $\begin{array}{l} (4.20) & u_{n} \to u weakly star in L^{\infty} (T, t; H_{0}^{2} (Ω)), \\ (4.21) & u_{n t} \to u_{t} weakly star in L^{\infty} (T, t; L^{2} (Ω)), \\ (4.22) & u_{n} \to u in L^{q + 1} (T, t; L^{q + 1} (Ω)), \\ (4.23) & u_{n} (t) \to u (t) and u_{n} (T) \to u (T) in L^{k} (Ω), \end{array}$ for $k < \frac{2 n}{n - 4}$ , where we use the compact embeddings $H_{0}^{2} (Ω) ↪ L^{2} (Ω)$ , $H_{0}^{2} (Ω) ↪ L^{k} (Ω)$ . In the following, we will deal with each term in (4.17) one by one.

Firstly, from the dissipativity, (4.20), (4.21) and (4.22), we obtain $\begin{array}{l} (4.24) & lim_{n \to \infty} lim_{m \to \infty} \int_{Ω} ε (t) (u_{n t} (t) - u_{m t} (t)) (u_{n} (t) - u_{m} (t)) d x = 0, \\ (4.25) & lim_{n \to \infty} lim_{m \to \infty} \int_{T}^{t} \int_{Ω} L (u_{n t} (s) - u_{m t} (s)) (u_{n} (s) - u_{m} (s)) d x d s = 0, \\ (4.26) & lim_{n \to \infty} lim_{m \to \infty} \int_{T}^{t} \int_{Ω} (f (u_{n} (s)) - f (u_{m} (s))) (u_{n} (s) - u_{m} (s)) d x d s = 0 . \end{array}$ Next, since $\begin{array}{l} \int_{T}^{t} \int_{Ω} (f (u_{n} (s)) - f (u_{m} (s))) (u_{n t} (s) - u_{m t} (s)) d x d s \\ = \int_{T}^{t} \int_{Ω} f (u_{n} (s)) u_{n t} (s) d x d s + \int_{T}^{t} \int_{Ω} f (u_{m} (s)) u_{m t} (s) d x d s \\ - \int_{T}^{t} \int_{Ω} f (u_{n} (s)) u_{m t} (s) d x d s - \int_{T}^{t} \int_{Ω} f (u_{m} (s)) u_{n t} (s) d x d s \\ = \int_{Ω} F (u_{n} (t)) d x - \int_{Ω} F (u_{n} (T)) d x + \int_{Ω} F (u_{m} (t)) d x - \int_{Ω} F (u_{m} (T)) d x \\ - \int_{T}^{t} \int_{Ω} f (u_{n} (s)) u_{m t} (s) d x d s - \int_{T}^{t} \int_{Ω} f (u_{m} (s)) u_{n t} (s) d x d s, \end{array}$ and combing (1.5) with (4.20), (4.21) and (4.23), taking first $m \to \infty$ , then $n \to \infty$ , we have $\begin{array}{l} lim_{n \to \infty} lim_{m \to \infty} \int_{T}^{t} \int_{Ω} (f (u_{n} (s)) - f (u_{m} (s))) (u_{n t} (s) - u_{m t} (s)) d x d s \\ = \int_{Ω} F (u (t)) d x - \int_{Ω} F (u (T)) d x + \int_{Ω} F (u (t)) d x - \int_{Ω} F (u (T)) d x \\ - \int_{T}^{t} \int_{Ω} f (u (s)) u_{t} (s) d x d s - \int_{T}^{t} \int_{Ω} f (u (s)) u_{t} (s) d x d s \\ = 0 . \end{array}$ Similarly, we have $\begin{array}{l} \int_{s}^{t} \int_{Ω} (f (u_{n} (ξ)) - f (u_{m} (ξ))) (u_{n t} (ξ) - u_{m t} (ξ)) d x d ξ \\ = \int_{Ω} F (u_{n} (t)) d x - \int_{Ω} F (u_{n} (s)) d x + \int_{Ω} F (u_{m} (t)) d x - \int_{Ω} F (u_{m} (s)) d x \\ (4.27) & - \int_{s}^{t} \int_{Ω} f (u_{n} (ξ)) u_{m t} (ξ) d x d ξ - \int_{s}^{t} \int_{Ω} f (u_{m} (ξ)) u_{n t} (ξ) d x d ξ . \end{array}$ At last, we deal with $\int_{T}^{t} \int_{s}^{t} \int_{Ω} (f (u_{n} (ξ)) - f (u_{m} (ξ))) (u_{n t} (ξ) - u_{m t} (ξ)) d x d ξ d s$ . Note that $| \int_{s}^{t} \int_{Ω} (f (u_{n} (ξ)) - f (u_{m} (ξ))) (u_{n t} (ξ) - u_{m t} (ξ)) d x d ξ |$ is bounded for each fixed t, then according to the Lebesgue dominated convergence theorem, we have $\begin{array}{l} lim_{n \to \infty} lim_{m \to \infty} \int_{T}^{t} \int_{s}^{t} \int_{Ω} (f (u_{n} (ξ)) - f (u_{m} (ξ))) (u_{n t} (ξ) - u_{m t} (ξ)) d x d ξ d s \\ (4.28) & = \int_{T}^{t} (lim_{n \to \infty} lim_{m \to \infty} \int_{s}^{t} \int_{Ω} (f (u_{n} (ξ)) - f (u_{m} (ξ))) (u_{n t} (ξ) - u_{m t} (ξ)) d x d ξ) d s = 0 . \end{array}$ Hence from (4.24)–(4.28), we obtain that $ϕ_{T}^{t} \in C (B_{t})$ and the sequence ${U (t, τ_{n}) x_{n}}_{n = 1}^{\infty}$ has a convergent subsequence. □

Theorem 4.3.

Under the assumptions ( 1.2 )–( 1.6 ), the process $U (t, τ) : H_{τ} \to H_{t}$ generated by the equation ( 1.1 ) has a invariant time-dependent global attractor ${A_{t}}_{t \in R}$ .

Proof.

From Theorem 3.1, Theorem 4.2 and Theorem 2.8, we know that there exists a unique time-dependent global attractor ${A_{t}}_{t \in R}$ . Moreover, thanks to the strong continuity of the process, we can obtain that ${A_{t}}_{t \in R}$ is invariant. □

Footnotes

Acknowledgement

This work was supported by the NSFC grants (12071192).

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