Time-dependent asymptotic behavior of the solution for evolution equation with linear memory

1.   Introduction

Let $\Omega\subset\mathbb{R}^3$ be a bounded domain with smooth boundary $\partial\Omega$. For any $\tau\in\mathbb{R},$ we consider the following equation:

where $u = u(x, t):\Omega\times(\tau, \infty)\rightarrow \mathbb{R}$ is an unknown function, and $u_{0}:\Omega\times(-\infty, \tau]\rightarrow \mathbb{R}$ is a given past history of $u$, $k(0), k(\infty) > 0$ and $k'(s) \leq 0$ for every $s \in \mathbb{R}^{+}$, $g(x)\in L^{2}(\Omega)$. $\varepsilon(t)\in C^{1}(\mathbb{R})$ is a decreasing bounded function with

especially, there exists a positive constant $L$ such that

The function $f\in C^{1}(\mathbb{R}), f(0) = 0$, satisfies the conditions

and

where $C$ is a positive constant, and $\lambda_{1}$ is the first eigenvalue of $A = -\triangle$ with Dirichlet boundary value condition.

Nonlinear evolution equations of this type arise as models of a vibration of a nonlinear elastic rod, which are used to represent the propagation of lengthwise-waves in nonlinear elastic rods and ion-sonic of space transformation by weak nonlinear effect; see for details ^[1,2,3].

Equation (1.1) becomes a strongly damped wave equation with a linear memory term when the coefficient function $\varepsilon(t)\equiv0,$ and it was discussed clearly in ^[4] and the references therein. When $\varepsilon(t)\equiv\varepsilon$, Eq (1.1) becomes an autonomous evolution equation, and the long-time behavior of the solutions can be well characterized by using the concept of global attractors under the framework of semigroups. In this case, when $\mu(s) = -k'(s)$ vanishes, Eq (1.1) reduces to the damped wave equation, which has been extensively discussed by many authors. For instance, Xie and Zhong ^[5,6] systematically investigated the existence of global attractors for (1.1) on weak and strong Hilbert spaces, respectively. Based on the global well-posedness results given in ^[7], Sun, Yang and Duan ^[8] constructed the uniformly asymptotic regularity of solution with respect to $\varepsilon\in[0, 1]$ for (1.1) when $g\in L^{2}(\Omega)$ and $g\in H^{-1}(\Omega)$, respectively, and they also obtained the existence of exponential attractors as well as the upper-semicontinuity of global attractors.

If $\varepsilon(t)$ is dependent on $t$, then Eq (1.1) becomes more complex and interesting. In this case, the long-time behavior of the solutions for (1.1) can be well characterized by the concept of time-dependent global attractors under the framework of processes, which have been discussed in ^{[9,10,11,12,13]}. Recently, Ma, Wang and Liu ^[14] investigated the existence and regularity of the time-dependent attractors for wave equations by using the operator decomposition technique along with compactness of translation theorem, also they proved the asymptotic structure as in ^[13]. In ^[15], they verified the asymptotic compactness of wave equations with nonlinear damping and linear memory by using the contractive functions method which was introduced in ^[10].

For our problem, by using the method of contractive functions ^[10], Liu and Ma ^[16] have obtained the existence of time-dependent global attractors of a nonlinear evolution equation with nonlinear damping and $\mu(s) = -k'(s) = 0$ in (1.1). For problem (1.1), we first introduce a new variable that is used to construct a relatively complicated triple solution space. Second, in order to prove compactness and regularity we use the decomposition technique as in ^[14]. Finally, we also prove the asymptotic structure of time-dependent global attractor as $\varepsilon(t) \rightarrow 0$ when $t\rightarrow \infty$.

The rest of this article is organized as follows: In the next section, we define some function set, and we recall some basic definitions and abstract results. In Section 3, the existence and regularity of time-dependent global attractor are obtained. Finally, in Section 4 we prove the asymptotic structure of time-dependent global attractor.

2.   Preliminaries

As in ^[4], we introduce the new variable

and differentiating the above equation, we get

with

For simplicity, we set $\mu(s) = -k'(s)$ and $k(\infty) = 1,$ where the memory component $\mu$ satisfies the following conditions:

Then, we can reformulate (1.1) as the following dynamical system:

with initial boundary conditions

where

and

Without loss of generality, set $H = L^{2}(\Omega)$ with inner product$\langle \cdot, \cdot\rangle$ and norm $\|\cdot\|$. For $s \in\mathbb{R}^{+}$ we define the hierarchy of (compactly) nested Hilbert spaces

Especially, we have the embedding $H^{s+1}\hookrightarrow H^{s}.$ Also, we denote $A = -\Delta$ with domain $D(A) = H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$.

For $s\in\mathbb{R}^{+}$, let $L^{2}_{\mu}(\mathbb{R}^{+}; H^{s})$ be the family of Hilbert spaces of functions $\varphi: \mathbb{R}^{+}\rightarrow H^{s}$, endowed with the inner product and norm, respectively,

We also need the spaces

Now, for $t\in\mathbb{R}$ and $s\in\mathbb{R}^{+}$, we introduce the following time-dependent spaces

with norms

where the space $H_t^{s+1}$ is endowed with the time-dependent norm $\|v\|^{2}_{s}+\varepsilon(t)\|v\|^{2}_{s+1}$.

The symbol is always omitted whenever zero. In particular, the time-dependent phase space where we settle the problem is

endowed with the time-dependent product norms

Now we recall some basic definitions and abstract results that will help us to get our main results.

Definition 2.1. ^[9,12] Let $\{X_{t}\}_{t\in\mathbb{R}}$ be a family of normed spaces. A process is a two parameter family of mappings ${U(t, \tau): X_{\tau}\rightarrow X_{t}, t\geq\tau, t, \tau\in\mathbb{R}}$ with properties

(i) $U(\tau, \tau) = Id$ is the identity operator on $X_{\tau}$, $\tau\in\mathbb{R}$;

(ii) $U(t, s)U(s, \tau) = U(t, \tau)$, $\forall t\geq s\geq \tau, \tau\in\mathbb{R}$.

For every $t\in\mathbb{R},$ let $X_{t}$ be a family of normed spaces, and we define the $R-$ball of $X_{t}$ as follows:

We denote the Hausdorff semi-distance of two nonempty sets $A, B \subset X_{t}$ by

Definition 2.2. ^[9,12] A family $\mathfrak{C} = \{C_{t}\}_{t\in\mathbb{R}}$ of bounded sets $C_{t}\subset X_{t}$ is called uniformly bounded if there exists $R > 0$ such that $C_{t}\subset \mathbb{B}_{t}(R), \; \; \forall t\in\mathbb{R}$.

Definition 2.3. ^[9,12] We say $\mathfrak{B} = \{B_{t}\}_{t\in\mathbb{R}}$ is a time-dependent absorbing set for the process $U(t, \tau)$, if $B_{t}\subset\mathbb{B}_{t}(R)$ is uniformly bounded and there exist $t_{0} = t_{0}(\mathfrak{C})\geq0$ such that

for every uniformly bounded family $\mathfrak{C} = \{C_{t}\}_{t\in\mathbb{R}}$.

Definition 2.4. ^[9,12] A family $\mathfrak{K} = \{K_{t}\}_{t\in\mathbb{R}}$ is called pullback attracting if it is uniformly bounded and

for every uniformly bounded family $\mathfrak{C} = \{C_{t}\}_{t\in\mathbb{R}}$.

Definition 2.5. ^[9,12] The time-dependent global attractor is the smallest family $\mathfrak{A} = \{A_{t}\}_{t\in\mathbb{R}}\in\mathbb{K}$, where $\mathbb{K} = \{\mathfrak{K} = \{K_{t}\}_{t\in\mathbb{R}}:K_{t}\subset X_{t}$ compact, $\mathfrak{K}$ pullback attracting$\},$ i.e. $A_{t}\subset K_{t}, \forall t\in\mathbb{R}$, for any element $\mathfrak{K} = \{K_{t}\}_{t\in\mathbb{R}}\in \mathbb{K}$.

Definition 2.6. ^[12] The process $U(t, \tau)$ is called

● closed if $U(t, \tau)$ is a closed map for any pair of fixed times $t\geq \tau$;

● T-closed for some $T > 0$ if $U(t, t-T)$ is a closed map for all $t$.

Definition 2.7. ^[12] We say that $\mathfrak{A} = \{A_{t}\}_{t\in\mathbb{R}}$ is invariant if

Remark 2.1. ^[12] If the time-dependent global attractor $\mathfrak{A}$ exists, and the process $U(t, \tau)$ is strongly continuous (or norm-to-weak continuous, or closed, or T-closed), then $\mathfrak{A}$ is invariant.

Theorem 2.1. ^[12] If $U(t, \tau)$ is asymptotically compact, then there exists a unique time-dependent attractor $\mathfrak{A} = \{A_{t}\}_{t\in\mathbb{R}}.$

If $U(t, \tau)$ is a T-closed process for some $T > 0$ and possesses a time-dependent global attractor $\mathfrak{A} = \{A_{t}\}_{t\in\mathbb{R}},$ then $\mathfrak{A}$ is invariant.

In order to prove the asymptotic structure of the time-dependent global attractors for the process $U(t, \tau),$ we recall some results from ^[13,14].

Here, we will focus on the case of a process $U(t, \tau)$ acting on a family of spaces $\{Z_{t}\}_{t\in\mathbb{R}}$ of the form

where $\mathcal{X}$ is a normed space, and $\{\mathcal{Y}_{t}\}_{t\in\mathbb{R}}$ is a family of normed space, endowed with the product norm

Let $\Pi_{t}:Z_{t} \rightarrow \mathcal{X}$ be the projection on the first component of $Z_{t}$, that is, $\Pi_{t}(x, y) = x$. Accordingly, if $C_{t} \subset Z_{t}$, then $\Pi_{t}C_{t} = \{x \in \mathcal{X}:(x, y)\in C_{t} \}$. If $\mathfrak{C} = \{C_{t}\}_{t\in\mathbb{R}}$, then $\Pi\mathfrak{C} = \{\Pi_{t}C_{t}\}_{t\in\mathbb{R}}$.

Definition 2.8. ^[13,14] Let $\mathfrak{A} = \{A_{t}\}_{t\in \mathbb{R}}$ be the time-dependent global attractor of $U(t, \tau)$. If $\mathfrak{A}$ is invariant, then $A_{t} = \{z(t)\in Z_{t}:z\; \; \mathrm{CBT}\; \; of\; \; U(t, \tau) \}$. Accordingly, we can write

where $z:t\mapsto z(t)\in Z_{t}$ is a complete bounded trajectory $\mathrm{CBT}$ of $U(t, \tau)$ if

Lemma 2.1. ^[13] Assume that, for any sequence $z_{n} = (x_n, y_{n})$ of a complete bounded trajectory (CBT) of the process $U(t, \tau)$ and any $t_{n}\rightarrow \infty$, there exists a complete bounded trajectory (CBT) $w$ of a semigroup $S(t)$ and $s\in\mathbb{R}$ for which

as $n\rightarrow \infty$ up to a subsequence. Then,

where $A_{\infty}$ is the global attractor in the phase space $\mathcal{X}$ for the autonomous system corresponding to the non-autonomous system with the coefficient $\varepsilon(t)\equiv0.$

Let $F(u) = \int^{u}_{0} f(s) ds$, and we can obtain the following lemma:

Lemma 2.2. ^[14] From dissipation condition (1.5), there exist two positive constants $k_{1}$ and $k_{2}$ and for some $0 < \nu < 1$ such that

Lemma 2.3. ^[14] Let $Y(t):[\tau, \infty)\rightarrow \mathbb{R}^{+}$ be an absolutely continuous function satisfying the inequality

for some $\epsilon > 0, k\geq0$ and where $h:[\tau, \infty)\rightarrow \mathbb{R}^{+}$ fulfills

with $m \geq 0$. Then,

Within this article, we often use Hölder and Young inequalities and denote positive constants by $C$, which will change in different lines or even in the same line.

3.   Existence of time-dependent global attractors

3.1.   Well-posedness and time-dependent absorbing set

In order to obtain the well-posedness of the solution associated with (2.4)–(2.5), we first make a priori estimates as follows:

Lemma 3.1. Assume that (1.2)–(1.5) and (2.2)–(2.3) hold, and then for any initial data $z_{\tau} = z(\tau) = (u_{0}, u_{1}, \eta_{0})\in \mathbb{B}_{\tau}(R_0)\subset\mathcal{H}_{\tau}$, there exists a constant $R > 0$, such that

Proof. Multiplying $(2.4)_{1}$ with $2u_{t}+2\delta u$ and integrating on $\Omega$, we find that

First, from condition (1.3), and by the Hölder, Young and Poincaré inequalities, there holds

where $\|u\|^{2}_{1}\geq \lambda_{1}\|u\|^{2}, \; \forall u\in H^{2}(\Omega)$.

Let

and

Then,

Integrating (3.4) from $\tau$ to $t$, we have

Next, we estimate (3.2) and (3.3), respectively. By using (1.3), (2.7) and the Hölder, Young, Poincaré inequalities, it follows that

Then,

Using (2.2), (2.3) there holds

and

Hence, from (2.6) and the condition (1.3), we get

where we have chosen $0 < \delta$ small enough such that

Let $M_{1} = \min\{\frac{\nu}{4}, \delta\}$, $M_{2} = \min\{\delta, \nu\delta, \frac{\rho}{2}\}$, $m_{1} = \frac{2}{\lambda_{1}\nu}\|g\|^{2}+k_{2}$, $m_{2} = \frac{1}{2\lambda_{1}}\|g\|^{2}+2\delta k_{1},$ and then from (3.5) we arrive at

Therefore, taking $K_{0} > \frac{m_{2}}{M_{2}}$, we have

As a result, if $(u, u_t, \eta)$ is the solution of the system, let $\mathcal{B}_{t} = \bigcup_{t\geq \tau}U(t, \tau)B_{\tau},$ where

Then, $\mathcal{B}_{t}$ is a bounded absorbing set for process $\{U(t, \tau)\}_{t\geq\tau}$.

On the other hand, from the above discussion, there exists a positive constant $R(R_0) > 0$ such that

The proof is completed. □

Lemma 3.2. Let the assumptions (1.2)–(1.5) and (2.2)–(2.3) hold, and then for any initial data $z_{\tau} = z(\tau) = (u_{0}, u_{1}, \eta_{0})\in \mathcal{H}_{\tau}$, on any interval $[\tau, t]$ with $t > \tau$, there exists a unique solution $(u(t), u_{t}(t), \eta^{t}(s))$ of the system (2.4)–(2.5) satisfying

Furthermore, let $z_{i}(\tau)\in\mathcal{H}_{\tau}$ be the initial data such that $\|z_{i}(\tau)\|_{\mathcal{H}_{\tau}}\leq R_0, (i = 1, 2)$, and $z_{i}(t)$ be the solution of problem(2.4)–(2.5). Then, there exists $\widetilde{C} = \widetilde{C}(R_0) > 0$, such that

Thus, the system (2.4)–(2.5) generates a strongly continuous process $U(t, \tau)$, where$U(t, \tau):\mathcal{H}_{\tau}\rightarrow \mathcal{H}_{t}$ acting as $U(t, \tau)z(\tau) = \{u(t), u_{t}(t), \eta^{t}(s)\}$, with the initial data $z_{\tau} = \{u_{0}, u_{1}, \eta_{0}\}\in\mathcal{H}_{\tau}$.

Proof. Based on Lemma 3.1, we can obtain the existence of a solution for problem (2.4)–(2.5) by using the Faedo-Galerkin approximation method, and the degenerate coefficient function $\varepsilon(t)$ in (2.4) is not causing a new difficult. See for details ^[5,12,17].

Consequently, we only need to verify the estimate (3.8). For this purpose, we assume that $z_{i}(t) = \{u_{i}(t), u_{it}(t), \eta^{t}_{i}(s)\} (i = 1, 2)$ are the solutions of (2.4)–(2.5) with the corresponding initial data $z_{i}(\tau) = \{u^{0}_{i}(\tau), u^{1}_{i}(\tau), \eta_{i}^{0}(s)\} (i = 1, 2),$ and there exists $R_{0} > 0$ such that $\|z_{i}(\tau)\|_{\mathcal{H}_{\tau}}\leq R_{0}, i = 1, 2$. □

According to Lemma 3.1 we ensure that

Let $\overline{z}(t) = \{\overline{u}(t), \overline{u}_{t}(t), \overline{\eta}^{t}(s)\} = U(t, \tau)z_{1}(\tau)- U(t, \tau)z_{2}(\tau),$ and then $\overline{z}(t)$ satisfies the following equation:

Taking the inner product of (3.10) with $2\overline{u}_{t}$ in $L^{2}(\Omega)$, we get

In line with (1.4), (3.9), Hölder inequality, Young inequality and embedding $H^{1}_{0}(\Omega)\hookrightarrow L^{6}(\Omega)$, it follows that

meanwhile, (2.2) and (2.3) mean

Together with (3.12) and (3.13), from (3.11) we deduce

So, according to the norm of (2.5), we can claim

where $\widetilde{C} = \max\{C_{R}, \rho\}.$ Thus, by using the Gronwall lemma with (3.14), we conclude the result (3.8).

Remark 3.1. Based on the argument, there exists $R$ such that $\mathfrak{B} = \{\mathbb{B}_{t }(R)\}_{t\in \mathbb{R}}$ is a time-dependent absorbing set for the process $\{U(t, \tau)\}_{t\geq \tau}$ associated with (2.4) and (2.5), and for $M_0(R_0) > 0$ there holds

Proof. Let $\delta\equiv 0$ in equality (3.4), and we get that

Integrating on $[\tau, t]$ and using inequality (3.6), we have $\int_{\tau}^\infty \|u_{t}(y)\|_1^{2}d y\leq M_0(>0).$ Then, together with Lemma 3.1, we conclude that (3.15) is true. □

3.2.   Asymptotic compactness and time-dependent global attractor

In this section, we do as in ^[14]. We find a suitable decomposition of the process, which is the sum of a decaying part and compact part. By a direct application of the abstract Theorem 2.1, we do this strategy to show that the process is asymptotically compact, and then the existence of the time-dependent global attractor is obtained.

For decomposition we write $f = f_{0}+f_{1}$, where $f_{0}, f_{1}\in C^{2}(\mathbb{R})$ satisfy

Let $\mathfrak{B} = \{\mathbb{B}_{t}(\mathrm{M}_{0})\}_{t\in\mathbb{R}}$ be a time-dependent absorbing set. Then, for any $z\in\mathbb{B}_{\tau}(\mathrm{M}_{0})$ and fixed $\tau\in\mathbb{R}$, we decompose the process $U(t, \tau)$ as follows:

where

solve respectively the systems

and

In the following lemma, the constant $C > 0$ depends only on $\mathfrak{B}$.

Lemma 3.3. If (1.2)–(1.5), (2.2)–(2.3) and (3.16)–(3.19) hold, then there exists $\delta = \delta(\mathfrak{B}) > 0$ such that

Proof. Repeating word by word the proof of Lemma 3.1 in the case of $U_{0}(t, \tau)$, we can get the bound

Multiplying Eq $(3.20)_{1}$ by $2v_{t}+2\delta v$ and integrating on $\Omega$, we find that

Define

where

Then, we get

From (3.17) and (3.23), we have

Therefore, by the same steps of the proof of Lemma 3.1, we deduce

Thus, combining with (3.26) and using the Gronwall lemma with the above, we complete the proof. □

Remark 3.2. Under the assumptions of Lemma 3.3, the following uniformly bounded holds:

Lemma 3.4. If (1.2)–(1.5), (2.2)–(2.3) and (3.16)–(3.19) hold, then there exists $M = M(\mathfrak{B}) > 0$ such that

where

Proof. Multiplying Eq $(3.21)_{1}$ by $2A^{\sigma}w_{t}+2\delta A^{\sigma}w$ and integrating it over $\Omega$, we get

where

Now, by using (1.3), (3.17), (3.27) and the embedding inequality $(\sigma < \frac{\sigma+1}{2})$, we have

Then, using the Hölder, Young inequalities, we get

Choose $\delta$ small enough and $C > 0$ large enough, and we can obtain

Hence, exploiting $(3.17)$, $(3.27)$ and some Sobolev embeddings $H^{1+\sigma} \hookrightarrow L^{\frac{6}{1-2\sigma}}$, $H^{1-\sigma} \hookrightarrow L^{\frac{6}{1+2\sigma}}$, and the continuous embedding $H^{\frac{(3p-6)}{2p}} \hookrightarrow L^{p}(\Omega)\; (p > 2)$, we have

Also, by using (3.16), we have

In addition, (2.2) and (2.3) mean

and

As a consequence, we can write (3.29) as

We can see that for $0 < \delta$ small enough,

According to (3.24) and taking $\delta$ small enough, we get

where $q(t) = C(\|u_{t}\|^{2}+\|v\|^{2}_{1}+\|v\|^{4}_{1})$. Remark 3.1 and Lemma 3.3 imply that

Now, applying Lemma 2.3, we get

Together with (3.22), the proof is completed. □

Especially, taking $\sigma = \frac{1}{3}$, we directly get

The proof is similar the above estimation, here we omit it.

Remark 3.3. In order to obtain a compact subset of $\mathcal{H}_{t}$, we also need the compactness of the memory term which is verified and proved in Lemma 3.6 in ^[14].

Theorem 3.1. Assume that (1.2)–(1.5), (2.2)–(2.3), (3.16)–(3.19) hold. The process $U(t, \tau):\mathcal{H}_{\tau}\rightarrow \mathcal{H}_{t}$ generated by problem (2.4)–(2.5) has an invarianttime-dependent global attractor $\mathfrak{A} = \{A_{t}\}_{t\in\mathbb{R}}$.

Proof. Denote the closure of $C_{t}$ in $L^{2}_{\mu}(\mathbb{R}^{+}, H^{1}_{0}(\Omega))$ by $\overline{C}_{t}$. According to Lemma 3.4 and Remark 3.3, we consider the family $\mathfrak{K} = \{K_{t}\}_{t\in\mathbb{R}}$, where

Applying the compact embedding $H^{\sigma+1}\times H_t^{\sigma+1}\hookrightarrow H^{1}_{0}(\Omega)\times H^{1}_{0}(\Omega)$, together with the compactness of $C_{t}$ in $L^{2}_{\mu}(\mathbb{R}^{+}, H^{1}_{0}(\Omega))$, we know that $K_{t}$ is compact in $\mathcal{H}_{t}$; since the injection constant $M$ is independent of $t$, the set $K$ is uniformly bounded. Finally, by Theorem 2.1 and Lemmas 3.1, 3.3 and 3.4, we conclude that there exists a unique time-dependent global attractor $\mathfrak{A} = \{A_{t}\}_{t\in\mathbb{R}},$ Furthermore, from the strong continuity of the process state in Lemma 3.2 and from Remark 2.1, the $\mathfrak{A}$ is invariant. □

3.3.   Regularity of time-dependent global attractor

The main result of this subsection is to prove $A_{t}$ is bounded in $\mathcal{H}_{t}^{1}$. Fix $\tau\in\mathbb{R}$, and for $z\in A_{\tau}$ we decompose again the process $U(t, \tau)z$ into the sum $U_{2}(t, \tau)z+U_{3}(t, \tau)z,$ where

solve respectively the systems

and

As a particular case of Lemma 3.3, we learn that

Lemma 3.5. If (1.2)–(1.5), (2.2)–(2.3), (3.16)–(3.19) hold, then there exists $M_{1} = M_{1}(\mathfrak{A}) > 0$ such that

Proof. Multiplying equation of $(3.33)_{1}$ by $2Aw_{t}+2\delta Aw$ and integrating it over $\Omega$, using $(3.33)_2$ we get

where

and

Choose $\delta > 0$ small enough and $C > 0$ large enough, and then we can obtain

By some calculation as in Lemma 3.4 and taking $\delta$ small enough, we deduce

We know from (3.31) that $A_{t}$ is bounded in $\mathcal{H}_{t}^{\frac{1}{3}}$. Consequently, exploiting some Sobolev embeddings $H^{\frac{(3p-6)}{2p}}\hookrightarrow L^{p} (p\geq 2)$, $H^{\frac{1}{3}}\hookrightarrow L^{\frac{18}{7}}(\Omega)$, $H^\frac{4}{3}\hookrightarrow L^{18}(\Omega)$, there holds

so

where $C$ depends on $\delta$, $L$. We finally get

and then applying the Gronwall lemma and calling (3.36) we get the result. □

Theorem 3.2. Assume that (1.2)–(1.5), (2.2)–(2.3), (3.16)–(3.19) hold. Then $A_{t}$ is bounded in $\mathcal{H}_{t}^{1}$, with a bound independent of $t$.

Proof. From (3.34) and Lemma 3.5, for all $t\in\mathbb{R},$ it yields

where

Since $\mathfrak{A}$ is invariant, this means

Hence, $A_{t}\subset\overline{K^{1}_{t}} = K^{1}_{t},$ and we get that $A_{t}$ is bounded in $\mathcal{H}_{t}^{1}$ with a bound independent of $t\in \mathbb{R}$. □

Lemma 3.6. For any $\tau\in \mathbb{R}, z = (u, u_t, \eta^{t})\in A_{t},$ there exists a positive constant $C,$ such that

Proof. Similar to the proof of Remark 3.1, we can easily get the result. □

4.   Asymptotic structure of the time-dependent attractor

In this section we investigate the relationship between the time-dependent global attractor of $U(t, \tau)$ for problem (2.4) and the global attractor of the limit equation formally corresponding to (2.4) when $t\rightarrow +\infty$. If $\varepsilon(t) = 0$ in (2.4), we can obtain the following wave equation:

Within our assumptions on Sections 1 and 2, it is well known that Eq (4.1) generates a strongly continuous semigroup $\{S(t)\}_{t \geq 0}$ acting on the space $H^{1}_{0}(\Omega)\times L^{2}(\Omega)\times L^{2}_{\mu}(\mathbb{R}^{+}, H^{1}_{0}(\Omega))$ associated with the problem (4.1), such that $S(t)\{u_{0}, u_{1}, \eta_{0}\} = \{u(t), u_{t}(t), \eta^{t}(s)\}$ is the solution of (4.1), where $\{u_{0}, u_{1}, \eta_{0}\}$ is the initial data of (4.1). Furthermore, $\{S(t)\}_{t \geq 0}$ admits the (classical) global attractor $A_{\infty}$ in the space of $H^{1}_{0}(\Omega)\times L^{2}(\Omega)\times L^{2}_{\mu}(\mathbb{R}^{+}, H^{1}_{0}(\Omega))$. See ^[4,17] for details.

Also, we know that, for any fixed $s\in \mathbb{R},$

where $\omega:\mathbb{R}\rightarrow H^{1}_{0}(\Omega)\times L^{2}(\Omega)\times L^{2}_{\mu}(\mathbb{R}^{+}, H^{1}_{0}(\Omega))$ is called a CBT of $S(t)$.

Next, we establish the asymptotic closeness of the time-dependent global attractor $\mathfrak{A} = \{A_{t}\}_{t\in \mathbb{R}}$ of the process generated by (2.4) and the global attractor $A_{\infty}$ of the semigroup $\{S(t)\}_{t \geq 0}$ generated by (4.1).

That is, we can obtain the following result.

Theorem 4.1. Under the assumptions (1.2)–(1.5), (2.2)–(2.3), (3.16)–(3.19), the following limits holds

To prove (4.2), we need to prove the following Lemma which is based on Lemma 4.1.

Lemma 4.1. For any sequence $z_{n} = (u_{n}, \partial_{t}u_{n}, \eta^{t}_{n})$ of CBT for the process $U(t, \tau)$ and any $t_{n}\rightarrow \infty$, there exists a CBT $y = (w, w_{t}, \xi^{t})$ of the semigroup $S(t)$ such that, for every $T > 0,$

and

as $n\rightarrow \infty$, up to a subsequence.

Proof. From (3.37), for every $T > 0$, $u_{n}(\cdot+t_{n})$ is bounded in $L^{\infty}(-T, T, H^{2})\cap W^{1, 2}(-T, T, H^{1}_{0}(\Omega)),$ and $\partial_{t}u_{n}(\cdot+t_{n})$ is bounded in $L^{\infty}(-T, T, H^{1})\cap L^{2}(-T, T, H^{2}_{0}(\Omega)) \cap W^{1, 2}(-T, T, H(\Omega)).$ Then by direct application of Corollary 5 in ^[18] show that ($u_{n}(\cdot+t_{n}), \partial_{t}u_{n}(\cdot+t_{n})$) is relatively compact in $C([-T, T], H^{1}_{0}(\Omega)\times L^{2}(\Omega))$. □

In addition, by Remark 3.3 and together with (3.37), we know that the sequence $\eta^{\cdot+t_{n}}_{t}(s)$ is bounded in the space $L^{\infty}(-T, T; L^{2}_{\mu}(\mathbb{R}^{+}, H^{2})\cap H^{1}_{\mu}(\mathbb{R}^{+}, H^{1}_{0}(\Omega)))$, so $\eta^{\cdot+ t_{n}}_{t}(s)$ is relatively compact in $C([-T, T], L^{2}_{\mu}(\mathbb{R}^{+}, H^{1}_{0}(\Omega)))$. Hence there exists a function

such that

hold.

In particular, $y = (w, w_{t}, \xi^{t}(s))\in C(\mathbb{R}\times\mathbb{R}\times(\mathbb{R}, \mathbb{R}^{+}), H^{1}_{0}(\Omega)\times L^{2}(\Omega)\times L^{2}_{\mu}(\mathbb{R}^{+}, H^{1}_{0}(\Omega)))$. Also, recalling (3.36), we have

We are left to show that $y$ solves (4.1). Define

then, we write Eq (2.4) of $(v_{n}(t), \partial_{t}v_{n}, \xi^{t}_{n}(s))$ in the form

We first prove that the sequence $\varepsilon_{n}(t)\triangle\partial_{tt}v_{n}$ converges to zero in the distributional sense. Indeed, for every fixed $T > 0$ and every smooth H-valued function $\varphi$ supported on $(-T, T)$, we have

Then, exploiting (3.37) again, we get

where the constant $C > 0$ also depends on $\varphi$. Since

we reach the desired conclusion

Now, taking into account (1.4), for every $T > 0$, we have the convergence

in the topology of $L^{\infty} (- T, T; H ^{-1})$ for every $T > 0$. At the same time, the convergences

hold (up to subsequence) in the distributional sense. Therefore, we end up with the equality

which together with (4.6), proves that $y(t)$ is a CBT of the semigroup $S(t)$.

Proof. Proof of Theorem 4.1. According to Lemma 4.1, for our problem, we can apply Lemma 2.1 with $\mathcal{X} = H^{1}_{0}(\Omega)\times L^{2}_{\mu}(\mathbb{R}^{+}, H^{1}_{0}(\Omega)), \mathcal{Y}_{t} = H^{1}_{0}(\Omega)$, the latter space endowed with the norm $\|\cdot\|_{\mathcal{Y}_{t}} = \sqrt{\varepsilon(t)}\|\cdot\|_{1}+\|\cdot\|$. Combining with Lemma 2.1, the result here should include the convergence of $u_t$ in the space $L^2(\Omega),$ namely, (4.4). Consequently, we complete the proof of Theorem 4.1. □

5.   Conclusions

Based on the theory of time-dependent attractor in time-dependent space, we discussed the asymptotic compactness for the nonlinear evolution equation with linear memory. By the method of operator decomposition, which overcoming the difficulty caused by the degenerate coefficient and memory term, and then the regularity and asymptotic structure of the time-dependent attractor are also proved, that means the combination for time-dependent attractor with the global attractor of the limit wave equation when the coefficient $\varepsilon(t)\rightarrow0$ as $t\rightarrow \infty$.

Acknowledgments

This work was supported partially by the National Natural Science Foundation of China (No. 11961059, 12101502).

Conflict of interest

The authors declare no conflict of interest.