Pullback attractors for the nonclassical diffusion equations with memory in time-dependent spaces

1.   Introduction

This paper concentrates on the following nonclassical diffusion equation:

and initial value

where the forcing term $g = g(x)\in L^{2}(\mathbb{R}^n)(n\geq 3)$ is known, $\lambda > 0$ is a constant and $\mathbb{R}^{\tau} = [\tau, \infty)$.

To study problem (1.1) with (1.2), we assume that the time-dependent perturbed parameter $\varepsilon(t)$, the nonlinearity $f$ and the hereditary memory $k(s)$ satisfy the following conditions, respectively:

$(\bf H1)$ The time-dependent perturbed parameter $\varepsilon(t)\in C^1(\mathbb{R})$ is a decreasing bounded function satisfying

and there exists a constant $L > 0$, such that

$(\bf H2)$ The hereditary memory kernel $k$ is a nonnegative summable function having the explicit form

and $\mu(s)\in L^1(\mathbb{R}^+)$ is a decreasing piecewise absolutely continuous in each interval $[0, T]$ with $T > 0$. In particular, $\mu(s)$ is allowed to exhibit (infinitely many) jumps. Moreover, we require that $\mu(0) < \infty$ and

holds for some $\theta > 0$ and almost every $s > 0$. As shown in ^[1,2], this is completely equivalent to the requirement that

for some $M\geq 1, \; \delta > 0$, every $r\geq 0$, and almost every $s > 0$, then we obtain

For convenience, we also assume

$(\bf H3)$ The nonlinearity $f\in C^1(\mathbb{R}^n\times\mathbb{R}, \mathbb{R})$ satisfies

and

where $\varphi(x) \in L^1(\mathbb{R}^n), \, f_0(x) = f(x, 0)\in L^2(\mathbb{R}^n)$ is given, $\alpha$, $l$ are two positive constants and $\alpha < \lambda$.

We denote by $F$ the function $F(x, s) = \int^{s}_{0}f(x, \upsilon)d\upsilon$, then,

Let $f_1(x, s) = f(x, s)+\alpha s$ and $F_1(x, s) = \int_{\tau}^sf_1(x, s)ds$, then,

and

Remark 1.1. The assumption of the nonlinearity comes from ^[3], but some constraints have been relaxed. For example, we will not require that $\varphi(x)$ is a nonnegative function and $l < \lambda$. The class of nonlinearities studied in ^[4,5,6] have a restriction on the upper growth by which an exponential nonlinearity (e.g., $f(u) = e^u$) does not hold ture.

According to the Dafermos' idea in ^[7], we also add a new variable $\eta^t$ to describe the past history of $u$, which is defined as follows:

Let $\eta^t_t = \frac{\partial}{\partial t}\eta^t, \; \eta^t_s = \frac{\partial}{\partial s}\eta^t$, then it gets

where $\mathbb{R}^{+} = [0, +\infty)$. Historical variable $u_{\tau}(\cdot, \tau-s)$ of $u$ satisfies

where $\Re > 0$ and $\sigma \leq \delta$ ($\delta$ is from (1.5)).

Thus the system (1.1) with (1.2) can be rewritten as follows:

and the initial data

From (1.13), it is easy to obtain the following estimate:

For Eq (1.1), it is often used to describe some physical phenomena, for example, non-Newtonian flows, soil mechanics and heat conduction theory. Specifically, when we study heat conduction problems in fluid mechanics or solid mechanics, if the influence of viscosity is emphasized, the classic heat-conduction equation is often extended to the following form (see e.g., ^[8,9,10]):

However, when we consider polymer and high viscous liquid, etc., some important factors such as the historical influence of $u$ and the disturbance coefficient of viscosity must be included ^[11], that is, the following evolution equation:

The asymptotic behavior of solutions of Eq (1.16) has been studied by many scholars (see e.g., ^{[3,12,13,14,15,16,17,18,19]} and the references therein). However, the research focused on the nonclassical diffusion equation with constant coefficient and bounded smooth domain early on, see e.g., ^{[20,21,22,23,24,25,26]}. In ^[11], the diffusion equation with memory was proposed in the study of heat conduction and relaxation of high viscosity liquids. The convolution term represents the influence of past history on its future evolution and describes more accurately the diffusive process in certain materials, such as high viscosity liquids at low temperatures and polymers. Hence, it is necessary and scientifically significant to study the nonclassical diffusion equation with the time-dependent coefficient (i.e., variable coefficient) and memory. Furthermore, for Eq $(1.16)$, we focus on the nonclassical diffusion equation with memory and a time-dependent perturbed parameter $\varepsilon(t)$ on a bounded domain. For example, for the case of $k(s) = 0$ in (1.1), in ^[27,28], the authors proved the existence and regularity of the time-dependent global attractors on time-dependent spaces when the nonlinearity satisfies $|f''(u)|\leq c(1+|u|)$ and critically exponential growth, respectively. In addition, Wang and Ma studied the existence, regularity and asymptotic structure of the time-dependent global attractors in ^[29] for this equation when $f$ meets polynomial growth of arbitrary order. In particular, Wang et al. ^[6] proved the existence of the time-dependent global attractors for the problem with the nonlinearity of critical exponential growth.

More recently, the fourth author of the article and other co-authors considered the autonomous and nonautonomous nonclassical diffusion equation with memory or without memory on bounded domains, see e.g., ^{[4,5,19,30,31,32]}. In these papers, the operator decomposition and contractive functional methods are used to obtain the asymptotic regularity of the solutions and verify asymptotic compactness. It is worth mentioning that Xie et al. ^[4] have obtained the existence of the time-dependent global attractors on bounded domain for the nonclassical diffusion equation (1.16), lacking instantaneous damping with the nonlinearity that satisfies the polynomial growth of arbitrary order. However, we focus on the unbounded domain. Therefore, we can see that there is few relevant studies for the asymptotic behavior of solutions of Eq (1.1) in time-dependent whole spaces under assumptions (H1)–(H3). This is because there are two major difficulties to obtain the existence of time-dependent global attractors.

$(i)$ First, because of the nonlinearity with no restriction on the upper growth, the higher asymptotic regularity of the solutions of Eq (1.1) can not be obtained using the method of ^[33,34].

$(ii)$ Second, due to the influence of the time-dependent perturbed parameter $\varepsilon(t)$ and the lacking of compact embedding theorem on unbounded domains $\mathbb{R}^n$, it is impossible to directly construct the contractive function to prove the asymptotic compactness for the corresponding process $\{U(t, \tau)\}_{t\geq \tau}$ of Eq (1.1) (see e.g., ^[29,30,35]).

For solving these problems, a new analytical technique combined with the operator decomposition method is used to obtain contractive function, and then the pullback asymptotic compactness for the process $\{U(t, \tau)\}_{t\geq \tau}$ of Eq (1.14) is proved. Furthermore, using this operator decomposition method, the asymptotic regularity of the solutions for Eq (1.14) is also proved. Then, the regularity of time-dependent global attractors for this equation on $\mathcal{H}_t^1\times L_{\mu}^2(\mathbb{R}^{+}; H^1(\mathbb{R}^n))$ ($\mathcal{H}_t^1$ is defined later) are established.

For conveniences, hereafter let $|\cdot |_p$ be the norm of $L^p(\mathbb{R}^n) \; (p\geq 1)$. Let $(\cdot, \cdot)$, $(\cdot, \cdot)+(\nabla\cdot, \nabla\cdot) = \langle\cdot, \cdot \rangle_{H^1(\mathbb{R}^n)}$ and $(\cdot, \cdot)+(\nabla\cdot, \nabla\cdot)+(\Delta\cdot, \Delta\cdot) = \langle\cdot, \cdot \rangle_{H^2(\mathbb{R}^n)}$ be the inner product of $L^2(\mathbb{R}^n)$, $H^1(\mathbb{R}^n)$ and $H^2(\mathbb{R}^n)$ respectively.

Let $\|\cdot\|_0, \|\cdot\|_1$ be the norm of $H^1(\mathbb{R}^n)$ and $H^2(\mathbb{R}^n)$ respectively, then,

The time-dependent space $\mathcal{H}^1_t = H^1(\mathbb{R}^n)$ and $\mathcal{H}_t^2 = H^2(\mathbb{R}^n)$ are equipped with the norms:

It is necessary to point out here that $\|\cdot\|^2_{\mathcal{H}^1_t}$ and $|\cdot|^2_2+\varepsilon(t)\|\cdot\|_0^2$ are equivalent. In fact the following inequality is obvious:

and

Denote the weight spaces $\mathcal{V}_0 = L_\mu ^2(\mathbb{R}^+; L^2(\mathbb{R}^n))$, $\mathcal{V}_1 = L_\mu ^2(\mathbb{R}^+; H^1(\mathbb{R}^n))$ and $\mathcal{V}_2 = L_\mu ^2(\mathbb{R}^+; H^2(\mathbb{R}^n))$, and their inner products and norms are defined as follows:

and

With the above notation, the phase spaces of Eq (1.14) can be denoted as

equipped the following norms:

respectively.

Particularly, we use $\{\mathcal{M}_t\}_{t\in\mathbb{R}}$ to denote a family of normed time-dependent spaces. Moreover, we introduce some common notations based on processes of time-dependent space(see e.g., ^{[35,36,37,38]}).

Let $\{\mathcal{M}_t\}_{t\in\mathbb{R}}$ be a family of normed time-dependent space. Note that the ball with radius of $R$ in $\mathcal{M}_t$ is

For any given $\varepsilon > 0$, we define the $\varepsilon$ neighborhood of set $B\subset \mathcal{M}_t$ as follows:

Hausdorff semidistance of between two nonempty sets $A, B\subset X_t$ is defined as

The plan of this paper is as follows. In Section 2, we recall some basic concepts as the time-dependent global attractors and useful results that will be used later. In Section 3, we first prove pullback asymptotic compactness of the process corresponding to problem (1.14) with (1.15) by constructing contractive function, and then we obtain the existence and the regularity of time-dependent global attractors to problem (1.14) with (1.15) in whole space $\mathbb{R}^n$.

2.   Preliminaries

In this section, we will recall some basic concepts of time-dependent global attractors and theories of the existence of time-dependent global attractors (see e.g., ^[35,36,37]).

Definition 2.1. Let $\{\mathcal{M}_t\}_{t\in\mathbb{R}}$ be a family of normed spaces. A two-parameter family of operators $U(t, \tau): \mathcal{M}_{\tau}\rightarrow \mathcal{M}_t$ is called a process if it satisfies the following properties:

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

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

Definition 2.2. A family of sets $\tilde{C} = \{C_t:C_t\subset \mathcal{M}_t \, \, is \, \, bounded \}_{t\in \mathbb{R}}$ is called uniformly bounded if there exists a constant $R > 0$, such that $C_t\subset \mathscr{B}_t(R)$ for any $t\in \mathbb{R}$.

Definition 2.3. A family of sets $\tilde{B} = \{B_t\}_{t\in \mathbb{R}}$ is called pullback absorbing if $\tilde{B} = \{B_t\}_{t\in \mathbb{R}}$ is uniformly bounded and for all $R > 0$, there exists a constant $t_0 = t_0(t, R)\leq t$ such that $U(t, \tau)\mathscr{B}_{\tau}(R)\subset B_t$ for any $\tau\leq t_0$.

The process $\{U(t, \tau)\}_{t\geq \tau}$ is called dissipative whenever it enters a pullback absorbing family $\tilde{B}_0 = \{B^0_t\}_{t\in \mathbb{R}}$.

Definition 2.4. A time-dependent absorbing set for the process $U(t, \tau)$ is a uniformly bounded family $\tilde{B} = \{B_t\}_{t\in \mathbb{R}}$ with the following characteristics: For any $R > 0$, there exists $t_0 = t_0(t, R)\geq 0$, such that

Definition 2.5. The process $U(t, \tau)$ is called pullback asymptotic compact if for any $t\in \mathbb{R}$, any bounded sequence $\{z_n\}_{n = 1}^{\infty}\subset \mathcal{M}_{\tau_n}$ and $\{\tau_n\}_{n = 1}^{\infty}\subset (-\infty, t]$ with $\tau_n\rightarrow {-\infty}$ as $n\rightarrow \infty$, the sequence $\{U(t, \tau_n)z_n\}_{n = 1}^{\infty}$ has a convergent subsequence in $\mathcal{M}_t$.

Definition 2.6. A time-dependent global attractor of the process $U(t, \tau)$ is the smallest family $\tilde{\mathscr{A}} = \{\mathscr{A}_t\}_{t\in \mathbb{R}}$ such that

(i) for every $t\in\mathbb{R}$, $\mathscr{A}_t$ is compact in $\mathcal{M}_t$;

(ii) $\tilde{\mathscr{A}}$ is pullback attracting, namely, $\tilde{\mathscr{A}}$ is uniformly bounded and

holds for any uniformly bounded family $\tilde{C} = \{C_{\tau}\}_{\tau\in \mathbb{R}}$ and every fixed $t\in \mathbb{R}$ and $\tau\leq t$.

Remark 2.1. The pullback attracting nature can be equivalently described in the light of pullback absorbing: A (uniformly bounded) family $\mathcal{K} = \{K_t\}_{t\in\mathbb{R}}$ is said to be pullback attracting if for any $\varepsilon > 0$ the family $\{\mathcal{O}^{\varepsilon}_t(K_t)\}_{t\in\mathbb{R}}$ is pullback absorbing.

Theorem 2.1. A time-dependent global attractor ${\tilde{\mathscr{A}}}$ exists and it is unique if and only if the process $U(t, \tau)$ is asymptotic compact, i.e., the set

is non-empty.

It can be seen from Definition 2.6 that the time-dependent global attractor is not necessarily invariant. This is mainly because that the process is not required to meet some continuity. If the process $U(t, \tau)$ satisfies the appropriate continuity, then the invariance of time-dependent global attractor ${\mathscr{tilde{A}}}$ can be obtained.

Definition 2.7. We say that $\tilde{\mathscr{A}} = \{\mathscr{A}_t\}_{t\in \mathbb{R}}$ is invariant if

Lemma 2.1. If the time-dependent global attractor $\tilde{\mathscr{A}}$ exists and the process $U(t, \tau)$ is a strongly continuous process, then $\tilde{\mathscr{A}}$ is invariant.

Next, we will state the definitions of contractive function and contractive process, which will be used to obtain asymptotic compactness of a family of process $\{U(t, \tau)\}_{t\geq \tau}$ (see e.g., ^{[23,35,39,40,41,42,43]}).

Definition 2.8. Let $\{\mathcal{M}_t\}_{t\in\mathbb{R}}$ be a family of Banach spaces and $\tilde{B} = \{B_t\subset \mathcal{M}_t\}_{t\in \mathbb{R}}$ be a family of uniformly bounded subset. We call function $\varphi(\cdot, \cdot)$, defined on $\mathcal{M}_{\tau}\times \mathcal{M}_{\tau}$, to be a contractive function on $B_{\tau}\times {B}_{\tau}$ if for any sequence $\{z_n\}_{n = 1}^{\infty}\subset B_{\tau}$, there exists a subsequence $\{z_{n_k}\}_{k = 1}^{\infty}\subset \{z_n\} _ {n = 1} ^ {\infty}$ such that

We use $\mathfrak{E}(B_{\tau})$ to denote the set all contractive function on $B_{\tau}\times B_{\tau}$.

Definition 2.9. Let $U(t, \tau)$ be a process on $\{\mathcal{M}_t\}_{t\in\mathbb{R}}$ and have a pullback bounded absorbing set $\tilde{B} = \{B_t\}_{t\in \mathbb{R}}$. $U(t, \tau)$ is called $\mathcal{M}_t$-contractive process if for any given $\varepsilon > 0$, there exist $T = T(\varepsilon)$ and $\varphi^t_{T}(\cdot, \cdot)\in \mathfrak{E}(B_T)$ such that

where $\varphi^t_{T}$ depends on $T$.

Next, we will give the method to prove the existence of time-dependent global attractors for evolution equations, which will be used in the later discussion.

Theorem 2.2. ^[35] Let $\{\mathcal{M}_t\}_{t\in\mathbb{R}}$ be a family of Banach spaces, then $U(t, \tau)$ has a time-dependent global attractor in $\{\mathcal{M}_t\}_{t\in\mathbb{R}}$, if the following conditions hold:

(i) $U(t, \tau)$ has a pullback absorbing set $\tilde{B} = \{B_t\}_{t\in \mathbb{R}}$ in $\{\mathcal{M}_t\}_{t\in\mathbb{R}}$;

(ii) $U(t, \tau)$ is a $\mathcal{M}_t$-contractive process.

Lemma 2.2. ^[44] Let $X\subset\subset H\subset Y$ be Banach spaces, with $X$ reflexive. Suppose that $\{u_n\}_{n = 0}^\infty$ is a sequence, uniformly bounded in $L^2(\tau, T; X)$ and $du_n/ dt$ is uniformly bounded in $L^p(\tau, T; Y)$, for some $p > 1$. Then, there is a subsequence of $\{u_n\}_{n = 0}^\infty$ that converges strongly in $L^2(\tau, T; H)$.

3.   Time-dependent global attractors in $\{\mathcal{M}_t\}_{t\in\mathbb{R}}$

In this section, we shall consider the existence of time-dependent global attractors in $\{\mathcal{M}_t\}_{t\in\mathbb{R}}$. For this purpose, we have to first discuss the well-posedness for Eq (1.14) with (1.15).

3.1.   The well-posedness of equation

The well-posedness for Eq (1.14) with (1.15) can be obtained by using Faedo-Galerkin method (see e.g., ^[38,45]). To this end, we first give the definition of weak solution.

Definition 3.1. For any $R > 0$ and $T > \tau$, let $I = [\tau, T]$, then the function $z = (u, \eta^t)$ defined on $\mathbb{R}^n\times I$ is called a weak solution of the problem (1.14) with initial value $z_\tau = (u_\tau, \eta^\tau)\in \mathscr{B}_\tau(R)\subset \mathcal{M}_\tau$ if

Furthermore, the following identity hold:

holds for any $(\omega, \varphi)\in C^{\infty}(I, \mathcal{H}_t^1)\times C^{\infty} (I, L_{\mu}^{2}(\mathbb{R}^+; H^1(\mathbb{R}^n)))$ and $a.e. \, t\in I$.

Lemma 3.1. For any $R > 0$, $T > \tau$ and $z_{\tau} = (u_{\tau}, \eta^{\tau})\in \mathscr{B}_\tau(R)\subset\mathcal{M}_{\tau}$, then the problem (1.14) with (1.15) has unique weak solution

which continuously depends on the initial data in $\mathcal{M}_\tau$, i.e., there exists a constant $\kappa > 0$ not related to $t$ such that the process $U(t, \tau)$ is Lipschitz continuous

By Lemma 3.1, we may define the process of solutions on time-dependent space $\{\mathcal{M}_t\}_{t\in\mathbb{R}}$:

In addition, it's easy to obtain that the process $U(t, \tau)$ is a strongly continuous process on the time-dependent phase space $\{\mathcal{M}_t\}_{t\in\mathbb{R}}$.

3.2.   Time-dependent absorbing sets

In the following discussion, let $C$ mean any positive constant and $Q(\cdot)$ be a monotonically increasing function on $[0, \, \infty)$ which may be different from line to line even in the same line.

We always assume that: The assumptions ${(\bf H1)}–{(\bf H3)}$ are true. Furthermore, $z(t) = (u(t), \eta^t)$ is a sufficiently regular solution of (1.14) with (1.15).

Lemma 3.2. For any $R > 0$ and $z_\tau = (u_\tau, \eta^\tau)\in \mathscr{B}_\tau(R)\subset \mathcal{M}_\tau$. Then, there exist positive two constants $\delta_1$ and $k_1$, such that

holds for any $\tau\leq t$.

Proof. Multiplying the first equation of (1.14) by $u$ in $L^2(\mathbb{R}^n)$ and the second one by $\eta^t$ in $\mathcal{V}_0$, then we obtain

and

Furthermore, we also get

and

From (3.5) and (3.6), one gets

Taking $\sigma = \frac{\lambda-\alpha}{8(1+\theta^2)}$ and let

Then combining (3.4) and (3.5), it follows that

Furthermore, let

Then we get

and

where $\delta_1 = \frac{1}{2a_0}\min\{\frac{1}{L}, \frac{1}{4\theta^2}, \frac{\sigma}{2}, \frac{\lambda-\alpha}{2}\}$.

Then from (3.9) and (3.12) we get

Applying Grönwall lemma, we have

By (3.11), it follows that

Let

Then the proof is complete.

Corollary 3.1. For any given $R\in\mathbb{R}^+$, let $z_{\tau} = (u_{\tau}, \eta^{\tau})\in\mathscr{B}_{\tau}(R) \subset\mathcal{M}_{\tau}$, then the process $U(t, \tau)$ corresponding to Eq (1.14) with (1.15) possesses a time-dependent bounded absorbing set $\tilde{B_0} = \{B^0_t = \mathscr{B}_{t}(\rho_0)\}_{t\in \mathbb{R}}$, that is there exists $t_0 = t_0(t, R) < t$, such that

In fact, let $t_0 = t_0(t, R) = t- \frac{1}{\delta_1}\ln\frac{Q(R)}{k_1} < t$ and $\rho_0 = 2k_1$, then the conclusion can be directly obtained from Lemma 3.2.

For brevity, later in this article, let $\tilde{B}^0$ be the bounded uniformly absorbing set obtained in Lemma 3.2, i.e.,

From Lemma 3.2, it follows that:

Corollary 3.2. For any $R > 0$ and $z_{\tau} = (u_{\tau}, \eta^{\tau})\in\mathscr{B}_{\tau}(R) \subset\mathcal{M}_{\tau}$, then there exists $\mathcal{K}_0 = \mathcal{K}_0(R)$ such that

holds for all $t-\tau \geq 0$.

Corollary 3.3. For any $R > 0$ and $z_{\tau} = (u_{\tau}, \eta^{\tau})\in\mathscr{B}_{\tau}(R) \subset\mathcal{M}_{\tau}$, then there exists $\rho_1 = \rho_1(k_1) > 0$, such that the following estimate:

holds for any $\tau\leq t$.

Proof. By (3.4), it's easy to obtain

Let $b_1 = \min\{1, \frac{\lambda-\alpha}{2}\}$ and integrating $(3.14)$ on $[t, t+1]$, we get

Let $\rho_1 = \frac{1}{b_1} \left(\frac{1}{2(\lambda-\beta_1)})|g|^2_2+|\varphi|_1\right)+\frac{1}{2b_1} k_1+|\varphi|_1$ and $\tau\leq t$, then the proof is complete.

Lemma 3.3. For any $R > 0$ and $z_\tau = (u_\tau, \eta^\tau)\in \mathscr{B}_\tau(R)\subset \mathcal{M}_\tau$. Then, there exist positive two constants $\delta_2$ and $k_2$, such that

holds for any $\tau < t$.

Proof. We now multiply the first equation of (1.14) by $u_t$ in $L^2(\mathbb{R}^n)$, then we get

Setting

From Lemma 3.2 and Corollary 3.2, we get that for any $t > \tau$, there exists $t^{\star}\in(\tau, t]$ such that

Furthermore, associating with (1.10), it is easy to obtain the following inequality:

holds for some $t^{\star}\in (\tau, t]$. Let

where

Then, $1-(1+|\lambda-l|)\beta > \frac{1}{2}, 1-2\beta > \frac{1}{2}$. Let

Therefore, we get

and

Combining with (3.4) and (3.15), it follows that

Let

Then,

Then, there is constant $\delta_2 > 0$, such that

Thus, we can rewrite (3.17) as follows:

Applying Grönwall lemma, we have

holds for any $\tau < t$. Then the proof is complete.

3.3.   Time-dependent global attractors

In subsection, we will prove the existence of time-dependent global attractors in $\{\mathcal{M}_t\}_{t\in\mathbb{R}}$ through the process $U(t, \tau)$ defined by (3.3). In order to prove Theorem 3.1, we first give the following lemmas.

Lemma 3.4. For any $R > 0$ and $z_{\tau} = (u_{\tau}, \eta^{\tau})\in\mathscr{B}_{\tau}(R) \subset\mathcal{M}_{\tau}$, then there exists a positive constant $\mathcal{K}_1 = \mathcal{K}_1(R)$, such that

holds for any $t\geq \tau$.

Proof. From (3.19) and the value of $\beta$, one gets

Integrating $t$ from $t$ to $t+1$ on both sides of (3.22), and organizing it to obtain

holds for any $\tau < t$. Let $\mathcal{K}_1 = Q(R)+C(|\varphi|_1+|g|^2_2+|f_0|_2^2)+a_1\beta|f_0|_2^2$, then the proof is complete.

Lemma 3.5. Let $B$ be any bounded subset $\mathcal{M}_{\tau}$ and $z_{\tau}\in B$. Then for any $\varepsilon > 0$, there exist positive constants $K_1$ large enough and $t_1\leq t$, such that

holds for every $\tau\leq t_1 \leq t$ and $k\geq K_1$, where $B_{k}^c = \{x\in\mathbb{R}^n: |x|\geq k\}$.

Proof. Let $\theta(\cdot)\in C^{\infty}(\mathbb{R}^+)$ satisfies

and

where $b > 0$ is a constant. Setting $\theta_k = \theta(\frac{2|x|^2}{k^2})$ and multiplying the first equation of (1.14) by $\theta_k^2u$ in $L^2(\mathbb{R}^n)$, then we obtain

where $E_k(t) = \frac{1}{2}\left[|\theta_ku|_2^2+\varepsilon(t)|\theta_k\nabla u|_2^2+|\theta_k\nabla\eta^t|_{\mu, 2}^2\right]$. Similar to (3.7), we define the function $N_k(t)$ as follows:

Then it is follows that

We can define a functional with a undetermined coefficient $\kappa$ as follows:

Let $0 < \kappa < \frac{1}{2\theta^2}$ be sufficiently small, then it follows that

and

Combining with (3.23), (3.25) and (3.26), we obtain

From (1.9), there is

Let $\delta_3 = \min\{\lambda-\alpha, \frac{1-\theta^2\kappa}{L}, \frac{\kappa}{2}\}$, then we have that

Next, we will estimate each item on the right side of (3.30). According to the definition of $\theta$, it can be seen that

Therefore,

Similarly, we also can obtain the following estimates:

Combining with (3.28), then (3.30) can be rewritten as follows:

By Grönwall lemma, it follows that

Combining with Lemma 3.3, we have

and for any $\varepsilon > 0$, there exists $K_1\geq 0$ large enough, such that for every $k\geq K_1$,

and let $t_1 = t -\frac{1}{\delta_3}\ln \frac{Q(R)}{\varepsilon}$, for every $\tau\leq t_1\leq t$, it follows that

By (3.27), then,

holds for every $\tau\leq t_1\leq t$ and $k\geq K_1$. So we have that

In order to obtain the asymptotic regularity estimates later, we decompose the solution $U(t, \tau)z_{\tau} = (u(t), \eta^t)$ into the following sum:

where $U_1(t, \tau)z_{\tau} = (v(t), \xi^t)$ and $K(t, \tau)z_{\tau} = (\omega(t), \zeta^t)$ solve the following equations respectively:

where $l$ is a constant from (1.5). It has an initial data

And

with the initial value conditions

Lemma 3.6. Assume that $K(t, \tau)z_{\tau} = (\omega(t), \zeta^t)$ is the solution of Eq (3.35) with (3.36). Then there exist positive constants $k_i(i = 3, 4)$, such that

and

hold for any $t-\tau > 0$.

Proof. This proof can imitate the proof of Lemma 3.2, Corollary 3.3 and Lemma 3.4 word by word, it should be noted that (3.17), (3.21) and $\|(\omega(\tau), \zeta^{\tau})\|_{\mathcal{M}_{\tau}} = 0$. Then hence is omitted.

Lemma 3.7. Assume that $U_1(t, \tau)z_{\tau} = (v(t), \xi^t)$ is the solution of Eq (3.33) with (3.34). Then,

holds for every $t\in \mathbb{R}^\tau$ fixed.

Proof. Multiplying the first equation of (3.33) by $v(t)$ and integrating in $L^2(\mathbb{R}^n)$ and the second one by $\xi^t$ in $\mathcal{V}$, we have

and

Furthermore,

Similar to (3.7), we introduce the functional

Then,

Setting up

Furthermore, we define the energy-like functional

Then we get

and

where $a_3 = \frac{1+\gamma\theta}{2}$ and $\tilde{a}_3 = \frac{\gamma}{2(1+\gamma)} > 0$.

Therefore, it follows that

By (3.41), we have

Let $\delta_4 = \frac{1}{4a_3}\min\{\frac{\gamma}{8}, \lambda-2\gamma-\gamma\theta^2, \frac{1-\gamma\theta^2}{L}\}$, then (3.45) can be rewritten as

Applying Grönwall lemma, we get that

Then, combining with (3.43) and (3.44), we have

is true for any $t\in \mathbb{R}^\tau$.

Lemma 3.8. There exists a positive constant $\mathcal{K}_3 = \mathcal{K}_3(R)$ which depends on $t$ and $\tau$, such that

holds for any $\tau \leq t\in\mathbb{R}$.

Proof. Acting on the first equation of (3.35) by $-\Delta\omega(t)$ on $L^2(\mathbb{R}^n)$, we have

Similar to (3.7), we introduce the functional

Then,

Therefore, from (3.48) and (3.49), it is that

By Corollary 3.2 and the Grönwall lemma, we have

holds for any $\tau\leq t \in \mathbb{R}$. Let

then it follows that

holds for any $\tau\leq t\in\mathbb{R}$.

Next, we will verify the existence and regularity of the pullback global attractors $\tilde{\mathscr{A}}$ for Eq (1.1).

Theorem 3.1. The family of process $U(t, \tau)$ for Eq (1.14) with initial conditions (1.15) is $\mathcal{M}_t$-contractive process on $B_T\in\tilde{B}^0$ (from (3.13)).

Proof. Let $z_i(t) = (u_i(t), \xi_i^t) = U(t, \tau)z_{\tau}^i \; (i = 1, 2)$ be the solutions to Eq (1.14) with the parameter $\varepsilon(t)$ and initial data $z^i_{\tau}\in B_{\tau}\in\tilde{B}^0\; (i = 1, 2)$ ($\tilde{B}^0$ is from (3.13)) respectively.

By (3.32), there is

It gets

and

Then for any $\varepsilon > 0$, there is $T = T(\varepsilon)\leq t$ such that

holds for any fixed $t\geq T$.

Let $(\varpi(t), \varsigma^t) = (\omega_1(t)-\omega_2(t), \zeta^t_1-\zeta^t_2)$ be the solution of the following system:

It has an initial value conditions

Next, we also define the functional

Then,

where $a_4 = \frac{1}{2}\max\{1+\frac{1}{2\theta}, \frac{\lambda(1+\theta)}{2(1+\theta^2)}\}$ and $\tilde{a}_4 = \frac{1}{2}\min\{1, \frac{\lambda}{2(1+\theta^2)}\}$.

Similar to (3.5)–(3.7), we get

and

Then it yields

Let $\delta_5 = \frac{1}{2 a_4}\min\{\frac{1}{L}, \frac{1}{4\theta^2}, \frac{\lambda}{8(1+\theta^2)}\}$, then it follows that

So, we get

And

Then setting

Combining with Lemmas 3.3 and 3.4. Applying Lemma 2.2, then the sequences $\{u_{n}(s)\}_{n = 1}^{\infty}$ is relatively compact in $L^2(T, t;L^2(B_{k}))$. In other words, for any sequences $\{z_{n}(T) = (u_{n}(T), \eta^{T}_n)\}\subset B_T \in\tilde{B}^0$, $\{z_{n}(t) = (u_{n}(t), \eta^t_n)\}$ is the solution of Eq (1.14) with the initial data $\{z_{n}(T) = (u_{n}(T), \eta^{T}_n)\}$ respectively. Then there exist subsequence $\{z_{n_k}\}\subset \{z_n\}$ satisfying

It follows that $\varphi^t_T\in \mathfrak{E}(B_T)$. Substituting (3.56) and (3.52) into (3.51), one gets

By Definitions 2.8 and 2.9, then $\varphi_{T}^{t}$ is contractive function in $B_T$. Therefore, it's easy to obtain that the process $U(t, \tau)$ is $\mathcal{M}_t$-contractive process on $B_T\in\tilde{B}^0$ (from (3.13)).

As the end of this article, we will deduce the main conclusion as the following theorem:

Theorem 3.2. The process $U(t, \tau)$ defined by (3.3) possesses a time-dependent global attractor $\mathscr{\tilde{A}}$ in $\{\mathcal{M}_t\}_{t\in\mathbb{R}}$, and ${\mathscr{\tilde{A}}}$ is non-empty, compact, invariant in $\{\mathcal{M}_t\}_{t\in\mathbb{R}}$ and pullback attracting in $\{\mathcal{M}_t\}_{t\in\mathbb{R}}$. Furthermore,

Proof. Thanks to Lemma 3.2 and Theorem 3.1, it's easy to get the existence of time-dependent global attractor $\mathscr{\tilde{A}}$ for the process $U(t, \tau)$ defined by (3.3) in time-dependent spaces $\{\mathcal{M}_t\}_{t\in\mathbb{R}}$. According to Lemmas 3.7 and 3.8, the pullback asymptotic regularity of the solutions of Eq (1.1) is proved, and the regularity of the time-dependent global attractor $\mathscr{\tilde{A}}$ is obtained. By Lemma 2.1 and (3.2), it follows that the invariance of time-dependent global attractor $\mathscr{\tilde{A}}$.

4.   Conclusions

We conclude the existence, uniqueness and regularity of time-dependent global attractors on whole space. The findings of this study can be considered as a supplement to our previous works, such as ^[4,5]. We overcome some essential difficulties for studying this kind of problem, including that the compact embedding is no longer valid under the case of unbounded domain and the nonlinear term fulfills the supcritical growth as well as the memory kernel satisfying more general assumptions. However, our results show that the method of operator decomposition that was proposed in ^[46] is available for dealing with the case of unbounded domain like (1.1).

Unfortunately, we fail to consider the existence of time-dependent global attractors for Eq (1.1) which lacks instantaneous damping on whole space, and further study the upper-semicontinuity of time-dependent global attractors between two kinds of equations. Future studies shall consider such issues using the ideas of the paper and ^[4,5], i.e., the asymptotic behavior of solutions for nonautonomous and autonomous equations (1.1) lacking instantaneous damping on whole space.

Use of AI tools declaration

The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The research was financially supported by National Natural Science Foundation of China (Nos. 11101053, 71471020) and Scientific Research Program Funds of NUDT (No. 22-ZZCX-016).

Conflict of interest

The authors declare no conflicts of interest.