Dynamics of a damped quintic wave equation with time-dependent coefficients

1.   Introduction

In this paper, we are concerned with the following semilinear damped wave model

Here, $\Omega\subset\mathbb{R}^{3}$ is a bounded domain with a smooth boundary, $A = -\Delta$, $g(u)$ is a given source term, and the coefficients $\alpha(t)$, $\beta(t)$, and $\gamma(t)$ all depend on time. From now on, we assume the external force $f\in L^{2}(\Omega)$.

The semilinear wave equations with time-dependent coefficients have been investigated quite extensively by several authors in recent years, with particular regard to its long-term behavior. For example, wave equations with time-dependent speed of propagation were investigated by Conti et al. ^[12,13], and very recently this model was generalized to the hyperbolic equations with time-dependent memory kernel in ^[15]; Uesaka et al. ^[19] made some significant progress in the oscillation property of semilinear wave equations with time-dependent coefficients, and subsequently the dynamics of this equation were studied by Aragão et al. in ^[1], including the continuity of pullback attractors.

It is worth observing that the nonlinear term $g$ satisfies $g(u)\sim u|u|^{q-1}$ with $1\leq q\leq3$ in the aforementioned papers. Thus, it seemed natural to extend these results to the sup-cubic case. The case of sup-cubic growth rate is a bit more complicated since the uniqueness of energy-weak solutions is unknown as $q > 3$, see, e.g., ^[3,20]. In order to overcome these difficulties, the authors in ^[4] studied the semilinear oscillon equation with the growth index $3 < q < 5$ by using parabolic approximations governed by the fractional powers of the wave operator. Another effective way to deal with sup-cubic nonlinearity is using S–S solutions, which have more delicate space-time integrability, such as $u\in L_{loc}^{4}(\mathbb{R}, L^{12}(\Omega))$. Very recently, based on the recent extension of Strichartz estimates for the bounded domains, the Eq (1.1) in the case $\alpha = \beta = 1$ with a sign changing damping and sub-quintic nonlinearity ($0 < q < 5$) was discussed in detail in ^[5].

In this paper, motivated by the studies in ^[1,5,14], we consider the problem of the existence of a uniform global attractor for Eq (1.1) with quintic nonlinearity in the natural energy phase space $H_{0}^{1}(\Omega)\times L^{2}(\Omega)$. The difficulties with this problem mainly stem from the following aspects:

● How do you generalize the usual Strichartz estimates to wave equations with time-dependent coefficients? In the case of the whole space $\Omega = \mathbb{R}^{3}$, the Strichartz type estimates for variable coefficient wave equations have been studied by many authors, one can refer to ^[16] and the references cited therein. However, when $\Omega\subset\mathbb{R}^{3}$ is a general bounded domain, as far as we known, the corresponding results are still lacking.

● How to establish the asymptotic compactness of the system generated by S–S solutions of Eq (1.1) with quintic nonlinearity? In the sub-quintic case, one can establish the so-called energy-to-Strichartz (ETS) estimate (2.2), and based on the ETS estimate, one can obtain the well-posedness, dissipativity, asymptotic compactness, and existence of attractors in way that is similar to the classical cubic case. In contrast to this, in the quintic case, the ETS estimate is only proved in the case when $\Omega = \mathbb{R}^{3}$ or $\Omega = \mathbb{T}^{3}$ with periodic boundary conditions. Since the ETS estimate for a general domain is still an open problem, it is impossible to deduce the asymptotic compactness by giving any control of the Strichartz norm in terms of the initial data, and the control of this norm may be a priori lost when passing to the limit $t\rightarrow \infty$, and the attractor may contain solutions that are less regular than the S–S ones, for which we may not have the energy equality.

In this paper, in order to circumvent the difficulties mentioned above, we present a new scheme to study the dynamics of the wave equations (1.1), and summarize the main method in Figure 1 for clarity.

(1) By rescaling techniques for time, we can reduce the Eq (1.1) to an equation of simple form (2.3), and then the well-posedness as well as the energy dissipativity of S–S solutions can be proved by the usual method.

(2) We apply a newly developed framework named evolutionary systems (see ^[7]) for studying the asymptotic dynamics of S–S solutions, and obtain the existence and structure of the uniform weak global attractor $\mathscr{A}_{w}$. Since the evolutionary systems $\mathfrak{E}$ (3.3) generated by S–S solutions may not be closed with respect to weak topology on the phase space, we follow an interesting technique initiated by Cheskidov and Lu in ^[10], which is based on taking a closure of the evolution systems $\bar{\mathfrak{E}}$ (3.4). In what follows, our main task is to show that $\mathfrak{E}((-\infty, \infty)) = \bar{\mathfrak{E}}((-\infty, \infty))$ via a newly developed approach presented in ^[20].

(3) Using the energy method developed in ^[3] and remembering the backward regularity of complete trajectories contained in $\mathfrak{E}((-\infty, \infty))$, we can prove that the uniform weak global attractor $\mathscr{A}_{w}$ constructed in Step 2 is in fact a strongly compact strong global attractor $\mathscr{A}_{s}$. Moreover, we obtain a bounded uniform attractor $\mathscr{A}_{s}^{\Sigma}\subset\mathscr{E}^{1}$ by a decomposition technique.

We impose the following standing assumptions on the nonlinear damping and coefficients:

Assumption 1.1. (G) $g\in \mathcal{C}^{2}({\mathbb{R}})$ with $g(0) = 0$ and satisfies

Here $3\leq q\leq5$, $G(s) = \int_{0}^{s}g(\tau)d\tau$, $\kappa_{i}$ ($i = 1, 2, 3, 4$), $\delta$, and $C_{g}$ are given positive constants.

(COEF) $\alpha, \beta, \gamma\in\mathcal{C}^{2}_{b}({\mathbb{R}})$ satisfying

where $\alpha_{i}$, $\beta_{i}$ and $\gamma_{i}$ ($i = 0, 1, 2$) are all positive constants, and $2\gamma_{0}\beta_{0} > \alpha_{2}\beta_{1}+\alpha_{1}\beta_{2}$.

The outline of our paper is given below. In Section 2, the property of the S–S solutions of Eq (1.1) is discussed in Theorem 2.2. In Section 3, the existence and structure of the uniform weak global attractor are studied in Theorem 3.7, and the backward asymptotic regularity of complete trajectories contained in $\bar{\mathfrak{E}}((-\infty, \infty))$ are proved in Theorem 3.11. Finally, the existence and regularity of the uniformly strong global attractor is established in Theorems 4.3 and 4.4.

2.   Well-posedness of S–S solutions

Let $\|\cdot\|$ and $\langle\cdot, \cdot\rangle$ be the usual norm and inner product in $L^{2}(\Omega)$. For convenience, we denote $\mathcal{H}^{s} = D(A^{\frac{s}{2}})$, $\mathscr{E}^{s} = \mathcal{H}^{s+1}\times\mathcal{H}^{s}$, $s\in\mathbb{R}.$ Then, $\mathcal{H}^{0} = L^{2}(\Omega)$, $\mathcal{H}^{1} = H_{0}^{1}(\Omega)$, $\mathcal{H}^{2} = H^{2}(\Omega)\cap H_{0}^{1}(\Omega)$, and $\mathcal{H}^{-1}$ is the dual space to $H_{0}^{1}(\Omega)$. In particular, we denote $\mathscr{E}: = \mathscr{E}^{0} = H_{0}^{1}(\Omega)\times L^{2}(\Omega)$.

Definition 2.1. A function $u(t)$ is a weak solution of Eq (1.1) iff $\xi_{u}: = (u, \partial_{t}u)\in L^{\infty}(\tau, T;\mathscr{E})$ and Eq (1.1) is satisfied in the sense of distribution, i.e.,

for any $\phi\in\mathcal{C}_{0}^{\infty}((\tau, T)\times\Omega)$. A weak solution is a Shatah–Struwe (S–S) solution of Eq (1.1) on the interval $[\tau, T]$ iff $u\in L^{4}(\tau, T;L^{12}(\Omega))$.

Theorem 2.2. Under Assumption 1.1, then for every $\xi_{u}(\tau) = (u_{0\tau}, u_{1\tau})\in\mathscr{E}$, the Eq (1.1) admits a unique global S–S solution $u(t)$ with the estimate

where the positive constant $\varpi$ and the monotone increasing function $\mathcal{Q}$ are independent of $u$, $t$ and $\tau$. In the sub-quintic case, we have, in addition, the estimate

Moreover, if $\xi_{u}(\tau)\in\mathscr{E}^{1}$, then Eq. (1.1) admits a unique global strong solution and estimate (2.1) also holds.

Proof. (Sketch) Using the change of variable $s = \phi(t) = \int_{0}^{t}\sqrt{\frac{\beta(\omega)}{\alpha(\omega)}}d\omega$ and chain rule (see [1, Section 3] for more details), we can rewrite the Eq (1.1) as follows:

where $u(x, t) = u(x, \phi^{-1}(s)) = v(x, s)$, $\eta(s) = \frac{(\sqrt{\alpha\beta})_{s}+\gamma}{\sqrt{\alpha\beta}}$ and $\mu = \phi(\tau)$. The local well-posedness of Eq (2.3) can be verified by using the Galerkin method and Strichartz estimate, and the global existence and regularity of the S–S solution can be proved by Morawetz–Pohozhaev identity and a prior estimate; see ^[17] for further details. Analyzing the term $\eta(s)$ and recalling the assumption (COEF), we have

Here, we can choose $\eta_{0} = \frac{2\beta_{0}\gamma_{0}-\alpha_{2}\beta_{1}-\alpha_{1}\beta_{2}}{2\beta_{1}\sqrt{\alpha_{1}\beta_{1}}}$ and $\eta_{1} = \frac{2\beta_{1}\gamma_{1}+\alpha_{2}\beta_{1}+\alpha_{1}\beta_{2}}{2\beta_{0}\sqrt{\alpha_{0}\beta_{0}}}$. Taking the multiplier $\partial_{s}v+\varepsilon v$ ($\varepsilon > 0$ is small enough) in Eq (2.3), and applying dissipative assumptions (1.3) and (1.4) and Gronwall's inequality, we can obtain the estimate (2.1) for $v$, e.g., see ^[5]. In the sub-quintic case, we can also obtain an estimate (2.2) for $v$ by using the standard bootstrapping method in ^[17]. Finally, in view of

then the theorem is completed. □

3.   Uniform weak global attractors

3.1.   Evolutionary systems

Here we recall some basic ideas and results from the abstract theory of evolutionary systems; see ^[7,8,9,10] for details. Let $(\mathcal{X}, d_{s}(\cdot, \cdot))$ be a metric space endowed with a metric $d_{s}$, which will be referred to as a strong metric. Let $d_{w}(\cdot, \cdot)$ be another metric on $\mathcal{X}$ satisfying the following conditions:

(1) $\mathcal{X}$ is $d_{w}$-compact.

(2) If $d_{s}(u_{n}, v_{n})\rightarrow0$ as $n\rightarrow \infty$ for some $u_{n}$, $v_{n}\in\mathcal{X}$, then $d_{w}(u_{n}, v_{n})\rightarrow0$.

Due to the property 2, $d_{w}(\cdot, \cdot)$ and $d_{s}(\cdot, \cdot)$ will be referred to as weak metric, and strong metric respectively. Let $\mathcal{C}([a, b];\mathcal{X}_{\bullet})$, where $\bullet = s$ or $w$, be the space of $d_{\bullet}$-continuous $\mathcal{X}$-valued functions on $[s, t]$ endowed with the metric

Let also $\mathcal{C}([a, \infty); \mathcal{X}_{\bullet})$ be the space of $d_{\bullet}$-continuous $\mathcal{X}$-valued functions on $[a, \infty)$ endowed with the metric

To define an evolutionary system, first let

and for each $I\in\mathcal{T}$, let $\mathfrak{F}(I)$ denote the set of all $\mathcal{X}$-valued functions on $I$.

Definition 3.1. A map $\mathfrak{E}$ that associates with each $I\in\mathcal{T}$ a subset $\mathfrak{E}(I)\subset\mathfrak{F}(I)$ will be called an evolutionary system if the following conditions are satisfied:

(1) $\mathfrak{E}([0, \infty))\neq\emptyset$.

(2) $\mathfrak{E}(I+s) = \{u(\cdot):u(\cdot-s)\in\mathfrak{E}(I)\}$ for all $s\in\mathbb{R}$.

(3) $\{u(\cdot)\mid_{I_{2}}:u(\cdot)\in\mathfrak{E}(I_{1})\}\subset\mathfrak{E}(I_{2})$ for all pairs $I_{1}$, $I_{2}\subset\mathcal{T}$, such that $I_{2}\subset I_{1}$.

(4) $\mathfrak{E}((-\infty, \infty)) = \{u(\cdot):u(\cdot)\mid_{[T, \infty)} \in\mathfrak{E}([T, \infty)), \forall T\in\mathbb{R}\}$.

We will refer to $\mathfrak{E}(I)$ as the set of all trajectories on the time interval $I$. Let $P(\mathcal{X})$ be the set of all subsets of $\mathcal{X}$. For every $t \geq 0$, define a map

Definition 3.2. A set $\mathscr{A}_{w} \subset \mathcal{X}$ is a d$_{w}$-global attractor of $\mathfrak{E}$ if $\mathscr{A}_{w}$ is a minimal set that is

(1) d$_{w}$-closed;

(2) d$_{w}$-attracting: for any $B \subset\mathcal{X}$ and $\epsilon > 0$, there exists $t_0$, such that

Definition 3.3. The $\omega_{\bullet}$-limit set $(\bullet = s, w)$ of a set $A \subset \mathcal{X}$ is

In order to extend the notion of invariance from a semiflow to an evolutionary system, we will need the following mapping:

Definition 3.4. A set $A \subset\mathcal{X}$ is positively invariant if

$A$ is invariant if

$A$ is quasi-invariant if, for every $a \in A$, there exists a complete trajectory $u \in$ $\mathfrak{E}((-\infty, \infty))$ with $u(0) = a$ and $u(t) \in A$ for all $t \in \mathbb{R}$.

As shown in ^[7,10], a semiflow $\{S(t)\}$ or a family of a processes $\{U_{\sigma}(t, \tau)\}$, $\sigma\in\Sigma$, defines an evolutionary system. In order to investigate the existence and structure of $\mathscr{A}_{w}$, we use a new method initiated by Cheskidov and Lu in ^[10] by taking a closure of the evolutionary system $\mathfrak{E}$. Let

Obviously, $\bar{\mathfrak{E}}$ is also an evolutionary system. We call $\bar{\mathfrak{E}}$ the closure of the evolutionary system $\mathfrak{E}$, and add the top-script $^{-}$ to the corresponding notations. Below is an important assumption that we will impose on $\mathfrak{E}$ in some cases.

$\diamond$ A1 $\mathfrak{E}([0, \infty))$ is pre-compact in $\mathcal{C}([0, \infty); \mathcal{X}_{w})$.

Theorem 3.5. ^[10] Let $\mathfrak{E}$ be an evolutionary system. Then the weak global attractor $\mathscr{A}_{w}$ exists. Furthermore, assume that $\mathfrak{E}$ satisfies A1. Let $\bar{\mathfrak{E}}$ be the closure of $\mathfrak{E}$. Then

(1) $\mathscr{A}_{w} = \omega_{w}(\mathcal{X}) = \bar{\omega}_{w}(\mathcal{X}) = \bar{\mathscr{A}}_{w} = \{u_{0}\in\mathcal{X}:u_{0} = u(0)\mathit{\text{ for some}\; }u\in\bar{\mathfrak{E}}((-\infty, \infty))\}$.

(2) $\mathscr{A}_{w}$ is the maximal invariant and maximal quasi-invariant set w.r.t. $\bar{\mathfrak{E}}$.

(3) (Weak uniform tracking property) For any $\epsilon > 0$, there exists $t_{0}$, such that for any $t^{*} > t_{0}$, every trajectory $u\in\mathfrak{E}([0, \infty))$ satisfies

for some complete trajectory $v\in\bar{\mathfrak{E}}((-\infty, \infty))$.

3.2.   The existence and properties of $\mathscr{A}_{w}$

Let $U_{\sigma}(t, \tau):\mathscr{E}\rightarrow \mathscr{E}$, $t\geq\tau$ be the S–S solution operator of Eq (1.1), where $\sigma = (\alpha, \beta, \gamma)\in\Sigma: = [T_{h}\sigma, h\in\mathbb{R}]_{\mathcal{C}_{b}^{2}(\mathbb{R})}$, $(T_{h}\sigma)(\cdot): = \sigma(\cdot+h)$, then $T_{h}\Sigma = \Sigma$. We construct the skew product flow by

where $\left(\xi, \sigma\right)\in \mathbb{E} = \mathscr{E}\times\Sigma$ with norm $\|(\xi, \sigma)\|_{ \mathbb{E}} = (\|\xi\|_{\mathscr{E}}^{2} +\|\sigma\|_{\mathcal{C}_{b}^{2}(\mathbb{R})}^{2})^{\frac{1}{2}}$. Then $\{\mathbb{S}(t)\}_{t\geq0}$ forms a semigroup. Now define an evolutionary systems (ES) on $\mathbb{E}$ by

where $\mathcal{X}: = \{\xi_{u}\in\mathscr{E}:\|\xi_{u}\|_{\mathscr{E}}^{2}\leq2\mathcal{Q}(\|f\|^{2})\}$. Let

where $\mathbb{X}_{w} = \mathcal{X}_{w}\times\mathcal{C}_{b}^{2}(\mathbb{R})$ and the metric on $\mathcal{C}([0, \infty); \mathbb{X}_{\bullet})$ defined in the same manner as (3.1).

Lemma 3.6. Suppose $\sigma$ is translation compact in $\mathcal{C}_{b}^{2}(\mathbb{R})$, and let $\xi_{u_{n}} = (u_{n}, \partial_{t}u_{n})$ be a sequence of S–S solutions of Eq (1.1) with symbols $\sigma_{n}$ such that $(\xi_{u_{n}}(t), \sigma_{n}(t))\in \mathbb{X}$ for all $t\geq t_{0}$. Then

Moreover, there exists a subsequence $n_{j}$ such that $\sigma_{n_{j}}$ converges in $\mathcal{C}_{b}^{2}(\mathbb{R})$ to some $\sigma\in\Sigma$ and $\xi_{u_{n_{j}}}$ converges to some $\xi_{u}$ in $\mathcal{C}([t_{0}, T];\mathscr{E}_{w})$, i.e., $(\xi_{u_{n_{j}}}, \phi)\rightarrow (\xi_{u}, \phi)$ uniformly on $[t_{0}, T]$ as $n_{j}\rightarrow \infty$ for all $\phi\in\mathscr{E}$.

Proof. Applying Theorem 2.2 and remembering that $\xi_{u_{n}}$ are the S–S solutions of Eq (2.3), thus we obtain (3.5). Now applying the Alaoglu compactness theorem to extract a subsequence $\xi_{u_{n_{j}}}$ which w$^{\ast}$-converges to some function $\xi_{u}\in L^{\infty}([t_{0}, T];\mathscr{E})$, i.e.,

Recalling the following compact embedding

for some $0 < \iota\leq 1$, we deduce that the weak-$\ast$ convergence (3.6) implies the strong convergence $\xi_{u_{n_{j}}}\rightarrow \xi_{u} \text{ in }\; \mathcal{C}([t_{0}, T];\mathscr{E}_{w})\}.$ The proof is completed. □

Theorem 3.7. Under Assumption 1.1 assume that $\sigma$ is translation compact in $\mathcal{C}_{b}^{2}(\mathbb{R})$. Then the uniform weak global attractor $\mathscr{A}_{w}$ for ES $\frak{E}$ defined by (3.3) exists. In addition, $\mathfrak{E}$ satisfies A1 and $\mathscr{A}_{w}: = \{(\xi_{u_{0}}, \sigma_{0}):(\xi_{u_{0}}, \sigma_{0}) = (\xi_{u}(0), \sigma(0))\mathit{\text{ for some}\; }\Xi = (\xi_{u}, \sigma)\in\bar{\mathfrak{E}}((-\infty, \infty))\}$. Moreover, $\forall\epsilon > 0$ there is some $t_{0}: = t_{0}(\epsilon)$ such that for any $t^{*} > t_{0}$ and every trajectory $\Xi\in\mathfrak{E}([0, +\infty))$ satisfies $d_{\mathcal{C}([0, \infty): \mathbb{X}_{w})}(\Xi, \Xi^{*}) < \epsilon$ for some complete trajectory $\Xi^{*}\in\bar{\mathfrak{E}}((-\infty, \infty))$.

Proof. The existence of the attractor $\mathscr{A}_{w}$ can be established by using Theorem 3.5 directly. Let $\Xi_{{n}}$ be a sequence in $\frak{E}([0, \infty))$. Using Lemma 3.6, we extract a subsequence (still denoting by $\Xi_{{n}}$) that converges to some $\Xi^{1}\in\mathcal{C}([0, 1]; \mathbb{X}_{w})$ as $n\rightarrow \infty.$ Passing to a subsequence and still denote $\xi_{u_{n}}$ once more, we obtain that $\Xi_{n}\rightarrow \Xi^{2}\in\mathcal{C}([0, 2]; \mathbb{X}_{w})$ as $n\rightarrow \infty$ for some $\Xi^{2}\in\mathcal{C}([0, 2]; \mathbb{X}_{w})$ with $\Xi^{1} = \Xi^{2}$ on $[0, 1]$. Continuing this diagonalization process, we get a subsequence $\Xi_{n_{j}}$ converges to $\Xi\in\mathcal{C}([0, \infty); \mathbb{X}_{w})$, and A1 is proven. The other statement contained in the above theorem can be proved by applying Theorem 3.5 again. □

Theorem 3.8. Under Assumption 1.1 assume that $\sigma$ is translation compact in $\mathcal{C}_{b}^{2}(\mathbb{R})$. Then the complete trajectory $\Xi = (\xi_{u}, \sigma)\in\bar{\mathfrak{E}}((-\infty, \infty))$ iff there exists a sequence of times $t_{n}\rightarrow -\infty$ and a sequence of S–S solutions $\xi_{u_{n}}(t)$ of Eq (1.1):

such that $(\xi_{u_{n}}, \sigma_{n})\rightharpoonup(\xi_{u}, \sigma)$ in $\mathcal{C}([-T, \infty); \mathbb{X}_{w})$ for any $T > 0$, where $\sigma_{n} = (\alpha_{n}, \beta_{n}, \gamma_{n})\in\Sigma.$

Proof. Let $\Xi = (\xi_{u}, \sigma)\in\bar{\mathfrak{E}}((-\infty, \infty))$ and denote $\Xi_{n} = \Xi|_{[t_{n}, \infty)}\in\bar{\mathfrak{E}}([t_{n}, \infty))$, where $t_{n}\rightarrow -\infty$ as $n\rightarrow \infty$. Obviously $\Xi_{n}\rightharpoonup\Xi$ in $\mathcal{C}([-T, \infty); \mathbb{X}_{w})$, $\forall T > 0$. Since $\Xi_{n}\in\bar{\mathfrak{E}}([t_{n}, \infty))$, then there exists a sequence $\{\Xi_{n}^{(k)}\}_{k = 1}^{\infty}\in\mathfrak{E}([t_{n}, \infty))$ such that $\Xi_{n}^{(k)}\rightharpoonup\Xi_{n}$ in $\mathcal{C}([t_{n}, \infty); \mathbb{X}_{w})$ as $k\rightarrow \infty.$ By a standard diagonalization process, we obtain that there exists a sequence $\Xi_{n}^{(n)}$ (denoted by $\Xi_{n} = (\xi_{u_{n}}, \sigma_{n})$) such that $\Xi_{n}\rightharpoonup\Xi$ in $\mathcal{C}([-T, \infty); \mathbb{X}_{w})$ for any $T > 0$. Recalling the definition of $\mathfrak{E}$ and $\Xi$, we know that $\xi_{u_{n}}$ is the S–S solution of Eq (1.1).

Conversely, let $\Xi_{n} = (\xi_{u_{n}}, \sigma_{n})\in\mathfrak{E}([t_{n}, \infty))$ and $\Xi_{n}\rightharpoonup\Xi$ in $\mathcal{C}([-T, \infty); \mathbb{X}_{w})$, $\forall T > 0$. So $\{\Xi_{n}|_{[-T, \infty)}:\Xi_{n}\in\mathfrak{E}([t_{n}, \infty)\}\subset\mathfrak{E}([-T, \infty))$ converges to $\Xi|_{[-T, \infty)}\in\mathcal{C}([-T, \infty); \mathcal{X}_{w})$. Thus $\Xi\in\bar{\mathfrak{E}}([-T, \infty))$ for any $T > 0$. By definition, this implies $\Xi\in\bar{\mathfrak{E}}((-\infty, \infty))$. □

Remark 3.9. Since every S–S solution $\xi_{u_{n}}$ can be obtained as a limit of Galerkin approximations (see ^[14,17] for more detail), then for any $\Xi = (\xi_{u}, \sigma)\in\bar{\mathfrak{E}}((-\infty, \infty))$, we can extract a sequence $\xi_{u_{k}}^{(k)}$ by using a standard diagonalization process again such that $\xi_{u_{k}}^{(k)}\rightharpoonup\xi_{u}$ in $\mathcal{C}([-T, \infty); \mathcal{X}_{w})$ for any $T > 0$, and $u_{k}^{(k)} = \sum_{l = 1}^{k}d_{l}^{k}(t)e_{l}$ satisfies

where $t_{k}\rightarrow -\infty$ as $k\rightarrow \infty$, $\{e_{k}\}_{i = 1}^{\infty}$ be the orthonormal system of eigenvectors of the Laplacian $-\Delta$ with Dirichlet boundary conditions and $\mathrm{P}_{k}$ is the projector from $L^{2}(\Omega)$ to $E_{k}: = span\{e_{1}, e_{2}, \cdots, e_{k}\}$.

Corollary 3.10. Let the assumptions of Theorem 3.8 be satisfied. Then, for any $\Xi = (\xi_{u}, \sigma)\in\bar{\mathfrak{E}}((-\infty, \infty))$, we have

for any $0 < \iota\leq1$, where $\mathcal{Q}(\cdot)$ is a monotone increasing function.

Proof. Let $\Xi = (\xi_{u}, \sigma)\in\bar{\mathfrak{E}}((-\infty, \infty))$, taking the multiplier $\partial_{t}v$ in (2.3) and combining Theorem 2.2 and Remark 3.9 to deduce that $\int_{-\infty}^{\infty}\|\partial_{t}u(r)\|^{2}dr\leq \mathcal{Q}(\|f\|^{2})$. In order to prove convergence in (3.9), we note that $\bar{\mathfrak{E}}((-\infty, \infty))$ is bounded in $\big(\mathcal{C}_{b}(\mathbb{R}; \mathscr{E})\cap\mathcal{C}_{b}^{1}(\mathbb{R}; \mathscr{E}^{-1})\big)\times\mathcal{C}_{b}^{2}(\mathbb{R})$ and $\sigma$ is translation compact in $\mathcal{C}_{b}^{2}(\mathbb{R})$, then the convergence is a standard corollary of dissipative integral in (3.9) and the compact embedding $[\mathcal{C}_{b}(\mathbb{R}; \mathscr{E})\cap\mathcal{C}_{b}^{1}(\mathbb{R}; \mathscr{E}^{-1})] \subset\subset\mathcal{C}_{loc}(\mathbb{R}; \mathcal{H}^{1-\iota}\times\mathcal{H}^{-\iota})$ for every $0 < \iota\leq1$. □

3.3.   Backward smoothness of complete trajectories

The following Theorem 3.11 discusses the backward smoothing property of the complete trajectory included in $\bar{\mathfrak{E}}((-\infty, \infty))$, and the proof is similar to the ones given in [20, Theorem 2.1], and for this reason we give a sketch of the main steps of the proof for the reader's convenience.

Theorem 3.11. Under Assumption 1.1 and assume that $\sigma$ is translation compact in $\mathcal{C}_{b}^{2}(\mathbb{R})$, then for every complete trajectory $\Xi = (\xi_{u}, \sigma)\in\bar{\mathfrak{E}}((-\infty, \infty))$, there exists a time $T = T(u, \sigma)$ such that $\xi_{u}\in\mathcal{C}_{b}((-\infty, T];\mathscr{E}^{1})$ and $\|\xi_{u}\|_{\mathcal{C}_{b}((-\infty, T];\mathscr{E}^{1})}\leq\mathcal{Q}(\|f\|^{2}, \|\sigma\|_{\mathcal{C}_{b}^{2}(\mathbb{R})})$.

Proof. We divide the proof into several steps.

Step 1. Rewrite Eq (2.3) as follows:

From the definition of $h$ and applying Theorem 2.2, we have $\|h(s)\|^{2}\leq\mathcal{Q}(\|f\|^{2})$ and $\int_{S}^{S+1}\|\partial_{s}h(s)\|_{\mathcal{H}^{2}}^{2}ds = \int_{S}^{S+1}\|\partial_{s}v(s)\|^{2}ds\leq\mathcal{Q}(\|f\|^{2}).$ Using Corollary 3.10, we infer that

Step 2. Applying Lemma 2.2 in ^[20], we know that for sufficiently large $L$ (depending on the coefficients in Assumption 1.1), the parabolic equation

possesses a unique solution $w(s)$ in the class $C_{b}(\mathbb{R}; \mathcal{H}^{2})$ with the following estimates:

and the following convergence

Step 3. For a sufficiently large $L$, there exists a time $S = S(v, L, \sigma)$ such that the problem

possesses a unique regular bounded backward solution $\xi_{z}\in\mathscr{E}^{1}$, which satisfies

To see this, let $z = w+Z$, where $w$ satisfying Eq (3.11), then $Z$ satisfies

We can apply the implicit function theorem in order to solve Eq (3.16) in the space

where $S$ is small enough. Applying Step 2, we have

Now, we intend to verify that the variation equation at $Z = 0$

is uniquely solvable for every $H\in L^{2}_{loc}((-\infty, S], \mathcal{H}^{1})$ such that

if $S$ is small enough. Firstly, taking the multiplier $\partial_{s}Z+\varepsilon Z$ in (3.18) yields

where

Choosing $L\geq4C^{2}\kappa_{1}^{2}$, we obtain

Combining (3.19) and (3.20) and recalling (3.12), there exists a sufficiently small parameter $\varepsilon > 0$ such that

and

Using (3.12), (3.13), and embedding $\mathcal{H}^{2}(\Omega)\subset L^{\infty}(\Omega)$, we have

Applying Gronwall's inequality, we deduce

Thus, the solution to (3.18) is unique. Secondly, taking the multiplier $-\Delta(\partial_{s}Z+\varepsilon Z)$ in (3.18), interpreting $g'(w)Z$ as an external force, and using (3.21) yields

Thus, the Eq (3.18) is uniquely solvable in space (3.17) if $S$ is small enough. Now, applying the implicit function theorem for Eq (3.16), for a sufficiently small $S\in\mathbb{R}$, there exists a unique solution $\xi_{Z}\in\Phi_{S}$ of problem (3.16) satisfying

Combining the estimates of $w$ in Step 2 and (3.23) we have

Finally, differentiate equation (3.14) and set $\partial_{s}z = \zeta$, we have

Recalling (3.10) and (3.24), we obtain

Similar to (3.22), we have

Since $\mathcal{H}^{2}\subset\mathcal{C}(\overline{\Omega})$, then (3.26) and (3.27) imply (3.15).

Step 4. We need to prove $z = v$, for $s\leq S$. Applying Remark 3.9, there exists a sequence of Galerkin approximations such that $\xi_{v_{k}}^{(k)}\rightharpoonup\xi_{v}$ in $\mathcal{C}([-T, \infty); \mathcal{X}_{w})$ for any $T > 0$, and

Here $s_{k} = \int_{0}^{t_{k}}\sqrt{\frac{\beta(\omega)}{\alpha(\omega)}}d\omega$, $\eta_{k}\in\mathcal{C}_{b}^{1}(\mathbb{R})$, so there exits a subsequence (still denote) $\eta_{k}\rightarrow \eta$ in $\mathcal{C}_{b}^{1}(\mathbb{R})$. Now let $z_{k}(s) = \mathrm{P}_{k}z(s)$, $s\leq S$. According to Step 3, the solution $\xi_{z}(s)$ is bounded in $\mathscr{E}^{1}$ when $s\leq S$, and consequently

Here we used the fact that $\mathcal{H}^{2}\subset\subset\mathcal{C}(\overline{\Omega})$ again, and the convergence of the Fourier series is uniform on compact sets. Now denote $V(s): = v(s)-z(s)$ and $V_{k}(s): = v_{k}^{(k)}(s)-z_{k}(s)$, recalling (3.28) yields the following equation:

where $A_{k}(s) = [\eta(s)-\eta_{k}(s)]\partial_{s}z_{k}(s)+\mathrm{P}_{k}[g(z(s))-g(z_{k}(s))]$. Taking the inner product between (3.30) and $\partial_{s}V_{k}+\varepsilon V_{k}$, we have

where

Using the fact that

we have

Combining (3.15), (3.29), and (3.31), we can infer that there exists a time $S'\leq S$ such that, for sufficiently large $k$, we have

Applying Gronwall's inequality, we obtain

where constants $C$ and $\varepsilon$ are independent of $k$. Indeed, we have

Now, passing to the limit $k\rightarrow \infty$ in (3.32), we obtain $\|\xi_{V}(s)\|_{\mathscr{E}} = 0$ for all $s\leq S'.$ The proof of Theorem 3.11 is now complete. □

Corollary 3.12. Let the assumptions of Theorem 3.11 be satisfied. Then the uniform weak global attractor $\mathscr{A}_{w}$ for ES $\frak{E}$ is in a more regular space: $\mathscr{A}_{w}\subset\mathscr{E}^{1}\times\mathcal{C}_{b}^{2}(\mathbb{R})$.

Proof. Let $\Xi_{u} = (\xi_{u}, \sigma)$ be the complete trajectory of Eq (1.1), then $\Xi_{v}: = (\xi_{v}, \eta)$ be the corresponding complete trajectory of Eq (2.3). Recall that there exists time $S_{0}$, such that $\xi_{v}(s)\in\mathscr{E}^{1}$ for all $s\leq S_{0}$. Due to Theorem 2.2, there is an extension $\bar{v}$ for $s\geq S_{0}$ such that $\bar{v}(s) = v(s)$ for $s\leq S_{0}$ and $\bar{v}(s)$ is a S–S solution of Eq (2.3) for all $s\in\mathbb{R}$. Indeed, we can conclude that $\xi_{\bar{v}}(s)\in\mathscr{E}^{1}$ for all $s\in\mathbb{R}$. We are now ready to prove that $\xi_{\bar{v}}(s) = \xi_{v}(s)$ for all $s\in\mathbb{R}$. Since $\Xi_{v}: = (\xi_{v}, \eta)\in\bar{\mathfrak{E}}((-\infty, \infty))$, and applying Remark 3.9, we get

where $s\geq s_{k}\text{ and }\lim\limits_{k\rightarrow \infty}s_{k} = -\infty$. Obviously, $\bar{v}_{k} = \mathrm{P}_{k}\bar{v}$ satisfying

Denote $W = v-\bar{v}$, $W_{k} = v_{k}^{(k)}-\bar{v}_{k}$, then combining (3.33) and (3.34) we know that $W_{k}$ satisfies

where

Taking the multiplier $\partial_{s}W_{k}+\varepsilon W_{k}$ in (3.35) yields

where

Choosing $L$ large enough and applying [20, Proposition 2.1], we discover that the right-hand side of (3.36) satisfies $\Lambda_{k}(s)\leq C\|B_{k}(s)\|^{2}$ and $C$ is independent of $k$. Invoking Gronwall's inequality, we have the estimate

Passing to the limit $k\rightarrow \infty$ and employing the convergence $B_{k}\rightarrow L(-\Delta)^{-1}W$ strongly in $\mathcal{C}((-\infty, S_{0}], \mathcal{H}^{0})$, $\|B_{k}\|_{\mathcal{C}((-\infty, S_{0}], \mathcal{H}^{0})}\leq C$ ($C$ is independent of $k$), $\mathcal{E}_{W_{k}}$ is equivalent to $\|\xi_{W_{k}}\|_{\mathscr{E}}^{2}$ and the fact that $\bar{v}(s) = v(s)$ for $s\leq S_{0}$, we have

Invoking again Gronwall's inequality to relation (3.38) and noting $v(S_{0}) = \bar{v}(S_{0})$, we derive that $v(s) = \bar{v}(s)$ for all $s\in\mathbb{R}$. □

Remark 3.13. The proof of Corollary 3.12 indicates that for any $\Xi = (\xi_{u}, \sigma)\in\bar{\mathfrak{E}}((-\infty, \infty))$, then $\xi_{u}$ is the S–S solution of Eq (1.1), i.e., $\bar{\mathfrak{E}}((-\infty, \infty)) = \mathfrak{E}((-\infty, \infty))$. Moreover, we have $\xi_{u}(t)\in\mathscr{E}^{1} \text{ for all }\; t\in\mathbb{R}.$

4.   Strong global attractors

We introduce some definitions; see ^[2,6,11] for more details.

Definition 4.1. Let $S(t)$ be a semigroup acting on a Banach space $\mathcal{Y}$. A set $\mathscr{A}_{s} \subset\mathcal{Y}$ is a (strong) global attractor of $S(t)$ if

(1) The set $\mathscr{A}_{s}$ is compact in $\mathcal{Y}$;

(2) The set $\mathscr{A}_{s}$ is strictly invariant: $S(t) \mathscr{A}_{s} = \mathscr{A}_{s}$;

(3) It is an attracting set for the semigroup $S(t)$, i.e., for any bounded set $B \subset\mathcal{Y}$,

Definition 4.2. A set $A \subset\mathcal{Y}$ is said to be uniformly (w.r.t. $\sigma\in\Sigma$) attracting for the family of processes $\left\{U_\sigma(t, \tau)\right\}, \sigma \in \Sigma$, if for any fixed $\tau \in \mathbb{R}$ and every bounded set $B\subset\mathcal{Y}$

A closed, uniformly attracting set $\mathscr{A}_{s}^{\Sigma}$ is said to be the uniform (w.r.t. $\sigma \in \Sigma$) attractor of the family of processes $\left\{U_\sigma(t, \tau)\right\}, \sigma \in \Sigma$, if it is contained in any closed uniformly attracting set (minimality property).

The kernel $\mathcal{K}_{\sigma}$ consists of all bounded complete trajectories of the process $U_{\sigma}(t, \tau)$, i.e.,

and $\mathcal{K}_{\sigma}(s)$ denotes the kernel section at a time moment $s\in\mathbb{R}$:

Theorem 4.3. Under Assumption 1.1 assume that $\sigma$ is translation compact in $\mathcal{C}_{b}^{2}(\mathbb{R})$. Then the semigroup $\mathbb{S}$ defined in (3.2) possesses a strong global attractor $\mathscr{A}_{s}$ in $\mathscr{E}^{1}\times\mathcal{C}_{b}^{2}(\mathbb{R})$, which coincides with the uniform weak attractor $\mathscr{A}_{w}$ in Theorem 3.7 and satisfies the following properties:

(i) $\Pi_1 \mathscr{A}_{s} = \mathscr{A}_{s}^{\Sigma}$ is the uniform (w.r.t. $\sigma \in \Sigma$) attractor of the family of processes $\left\{U_\sigma(t, \tau)\right\}$, $\sigma\in\Sigma$ and $\Pi_{1}$ is projector from $\mathscr{E} \times \Sigma$ onto $\mathscr{E}$;

(ii) The uniform attractor satisfies $\mathscr{A}_{s}^{\Sigma} = \bigcup\limits_{\sigma \in \Sigma} \mathcal{K}_\sigma(0)$, where $\mathcal{K}_\sigma(0)$ is the section at $t = 0$ of the kernel $\mathcal{K}_\sigma$ of the process $\left\{U_\sigma(t, \tau)\right\}$ with the symbol $\sigma \in \Sigma$.

Proof. In order to apply [6, Theorem Ⅳ.5.1] and [18, Theorem 3.4], we have to check that the processes $\{U_{\sigma}(t, \tau)\}_{\sigma\in\Sigma, t\geq\tau}$ corresponding to the S–S solutions of Eq (1.1) be $(\mathscr{E}\times\Sigma, \mathscr{E})$-continuous and uniformly asymptotically compact.

Firstly, consider two S–S solutions $\xi_{u_{i}}$ of Eq (1.1) with symbols $\sigma_{i}$ and with initial values $\xi_{u_{i\tau}}$, then correspondingly, $\xi_{v_{i}}$ are S–S solutions of Eq (2.3) with symbols $\eta_{i}$, $i = 1, 2$. Then $\xi_{w} = \xi_{v_{1}}-\xi_{v_{2}}$ satisfies the equation

Taking the scalar product of (4.1) with $\partial_{s}w$, we obtain

Thanks to Assumption 1.1 (G) and applying the elementary inequality, we have

Where $q(s) = (1+\|v_{1}(s)\|_{L^{12}(\Omega)}^{4}+\|v_{2}(s)\|_{L^{12}(\Omega)}^{4})$. Applying (4.3) and (4.4) in (4.2) and employing Gronwall's inequality, we obtain that

Then the $(\mathscr{E}\times\Sigma, \mathscr{E})$-continuity follows in a standard way from the energy inequality (4.5).

Secondly, we intend to verify the uniform asymptotic compactness of the processes $\{\widetilde{U}_{\eta}(s, \mu)\}_{\eta\in\widetilde{\Sigma}, s\geq\mu}$ corresponding to the S–S solutions of Eq (2.3), where $\widetilde{\Sigma} = [T_{h}(\eta), h\in\mathbb{R}]_{\mathcal{C}_{b}^{1}(\mathbb{R})}$. Let $\{\eta_{n}\}\subset\widetilde{\Sigma}$, $\{-\mu_{n}\}\subset(-\infty, 0]$, $\mu_{n}\rightarrow \infty$ as $n\rightarrow \infty$. $\xi_{\mu_{n}}$ belongs to a bounded subset in $\mathscr{E}$. Since $\sigma$ is translation compact in $\mathcal{C}_{b}^{2}(\mathbb{R})$, without loss of generality, we may assume that $\eta_{n}\rightarrow \eta$ $(n\rightarrow \infty)$ in $\mathcal{C}_{b}^{1}(\mathbb{R})$, and $\xi_{\mu_{n}}\rightharpoonup\xi_{\mu}$ weakly in $\mathscr{E}$ as $n\rightarrow \infty$. Denote $\xi_{v_{n}}(s) = \widetilde{U}_{\eta_{n}}(s, \mu_{n})\xi_{\mu_{n}}$ the corresponding solutions, then $v_{n}$ solves

Taking the multiplier $\partial_{s}v_{n}+\varepsilon v_{n}$ with $0 < \varepsilon\ll1$ in Eq (4.6), we derive the following energy type identity:

where

Now, integrate Eq (4.7) with respect to $s\in[-\mu_{n}, 0]$ to deduce that

Our intention now is to pass to the limit $n\rightarrow \infty$ in equality (4.8). To do this, we remind that $\xi_{v_{n}}$ is uniformly bounded in $\mathcal{C}((-\mu_{n}, \infty], \mathscr{E})$ and $\eta_{n}\rightarrow \eta$ $(n\rightarrow \infty)$ in $\mathcal{C}_{b}^{1}(\mathbb{R})$, then we get that

and $\Xi = (\xi_{v}, \eta)\in\bar{\mathfrak{E}}((-\infty, \infty)) = \mathfrak{E}((-\infty, \infty))$ by recalling Corollary 3.12. In addition, we also know that $\xi_{v}$ is an S–S solution and $\xi_{v_{n}}(0)\rightharpoonup\xi_{v}(0)$ weakly in $\mathscr{E}$. Applying the compact embedding $\mathcal{C}_{loc}((-\infty, 0], \mathscr{E})\subset\subset\mathcal{C}_{loc}((-\infty, 0], \mathcal{H})$, we can get that $v_{n}\rightarrow v$ strongly in $\mathcal{C}_{loc}((-\infty, 0], \mathcal{H})$, including almost everywhere. On the other hand, from the assumption (1.3), we can choose $\varrho = \frac{\varepsilon}{4}$, which guarantees that $\mathcal{G}_{v_{n}}(s)\geq -\kappa_{2}|\Omega|$ and choose $0 < \varepsilon\leq\varepsilon_{0}$ small enough such that the quadratic form $\mathcal{Q}_{v_{n}}$ is positive definite and satisfying $C_{1}\|\xi_{v}\|_{\mathscr{E}}^{2}\leq\mathcal{Q}_{v}\leq C_{2}\|\xi_{v}\|_{\mathscr{E}}^{2}$. Now, using the Fatou lemma, we conclude that

According to Theorem 3.11, $v$ is an S–S solution with more regularity in $\mathscr{E}^{1}$, and obviously $v$ satisfies the energy equality. Then, by repeating the derivation of (4.8), for solution $v$, we obtain the energy equality

Combining (4.9) and (4.10), we ascertain

Applying the Fatou lemma and weak lower semi-continuous of the norm, we find that

Obviously, energy equality is true only when inequalities (4.12) are also equalities. Recalling $\xi_{v_{n}}(0)\rightharpoonup\xi_{v}(0)$, we obtain $U_{\eta_{n}}(0, \mu_{n})\xi_{\mu_{n}} = \xi_{v_{n}}(0)\rightarrow \xi_{v}(0)$ strongly in $\mathscr{E}$. Finally, we get the uniformly asymptotically compact of the processes $\{U_{\sigma}(t, \tau)\}_{\sigma\in\Sigma, t\geq\tau}$ and the theorem is proved. □

Theorem 4.4. Under Assumption 1.1 assume that $\sigma$ is translation compact in $\mathcal{C}_{b}^{2}(\mathbb{R})$. Then the global attractor $\mathscr{A}_{s}$ of the solution semigroup $\mathbb{S}(t)$ defined in (3.2) is a bounded set in $\mathscr{E}^{1}\times\mathcal{C}_{b}^{2}(\mathbb{R})$.

Proof. For any initial data $(\xi_{u_{0}}, \sigma_{0})\in\mathscr{A}_{s}$, we will prove that

where the positive constant $\lambda$ and monotone increasing function $\mathcal{Q}(\cdot)$ are independent of $u$ and $t$. We proceed in three steps:

Step 1. Claim #1: Denote the restriction of the trajectory in $\mathfrak{E}((-\infty, \infty))$ to the time interval $t\in[0, 1]$ as

Then $\mathcal{F}$ is a compact set of $L^{4}(0, 1;L^{12}(\Omega))$:

Proof of claim. First note that the attractor $\mathscr{A}_{s}$ is compact in $\mathscr{E}\times\mathcal{C}_{b}^{2}(\mathbb{R})$, then there exists $T = T(\mathscr{A}_{s}) > 0$ such that

for any S–S solution $u(t)$ with $\xi_{u}(0)\in\Pi_{1}\mathscr{A}_{s}$, where $C$ may depend on $\mathscr{A}_{s}$, but is independent of $u$. Indeed, we obtain

Using $\mathfrak{E}((-\infty, \infty))$ is invariant with respect to time shifts, for any $\Xi = (\xi_{u}, \sigma)\in\mathfrak{E}((-\infty, \infty))$ we have

for some positive constant $C_{0}$, independent of $u$. Now let $\xi_{v_{i}}$ are two S–S solutions of Eq (2.3) with symbols $\eta_{i}$, $i = 1, 2$, and denote $\xi_{w} = \xi_{v_{1}}-\xi_{v_{2}}$, then recalling (4.5) and applying (4.15) or (4.16) to find that

where the constants $C$ and $K$ are independent of $\xi_{v_{i}}(0)$, $i = 1, 2$. Then, applying (1.2) and (4.17), we have

Applying Strichartz estimates for Eq (4.1) and recalling (4.17) and (4.18) gives us

Assertion (4.14) is now a consequence of (4.19) and the general fact that $\mathscr{A}_{s}$ is compact in $\mathscr{E}\times\mathcal{C}_{b}^{2}(\mathbb{R})$.

Step 2. Claim #2: For any $\varepsilon > 0$ and any $\Xi = (\xi_{u}, \sigma)\in\mathfrak{E}((-\infty, \infty))$, we can split the solution into two parts $u = \bar{u}+\tilde{u}$, where

Here, the constant $C_{\varepsilon}$ depends on $\varepsilon$, but is independent of $u$.

Proof of claim. In fact, $\forall\varepsilon > 0$, there exists a finite $\varepsilon$-net $\{(y_{i}, \sigma_{i})\}_{i = 1}^{m}\subset\mathcal{D}$ satisfying

where $\mathcal{D}: = \{(u, \sigma)|_{t\in[0, 1]}, \Xi = (\xi_{u}, \sigma)\in\mathfrak{E}((-\infty, \infty))\}$ and $B_X\left(x_0, r\right)$ denotes the $r$-ball centered on $x_0$ in the space $X$. By (4.16), we also have

Then approximate $y_i$ by a smoother function $\tilde{y}_{i}$ such that

where $i = 1, 2, \cdots, m$ and the constant $C_{\varepsilon}$ is independent of $y_i$. Combining (4.21) and (4.22), we otain

For every $\Xi = (\xi_{u}, \sigma)\in\mathfrak{E}((-\infty, \infty))$, we observe that

So in view of (4.23) and (4.24), there exists $(\tilde{y}_{i_{n}}, \sigma_{i_{n}})$ such that

Define the function $\tilde{u}(t)$ as

and the function $\tilde{u}(t) = u(t)-\bar{u}(t)$. Then

For any $t\geq0$, $[t, t+1]\subset[n, n+2)$ for some $n$, then combining (4.24) and (4.25) leads us to the estimate

Hence, (4.26) and (4.27) imply the stated assertion (4.20) easily.

Step 3. The following estimates will be deduced by a formal argument, which can be justified by using Faedo–Galerkin method. Differentiating Eq (2.3) and setting $\theta(s) = \partial_{s}v$, we see that

with the initial condition

Taking the multiplier $\partial_{s}\theta+\lambda\theta$ in (4.28), we can discover

where

Choosing $\lambda$ small enough such that

We employ the decomposition (4.20) and then let

to discover

where $l_{\varepsilon}(s) = C(1+\|\tilde{v}\|_{L^{12}(\Omega)}^{3}+\|\bar{v}\|_{L^{12}(\Omega)}^{3})\|\tilde{v}\|_{L^{12}(\Omega)}$. Owing to (4.16) and (4.20), we conclude

for some positive constant C independent of $\varepsilon$. Combining now (4.30) and (4.31) and employing Gronwall's inequality, we deduce

for some monotone function $\mathcal{Q}(\cdot)$ and positive constant $\lambda$, which are independent of $\varepsilon$ and $v$. We estimate using (4.32) and (4.33):

Recalling now (4.29), we see that in fact

Inserting this estimate into (4.34), we discover that

Recalling (2.3) and employing (1.2), we deduce that

Combining (4.35) and (4.36) and remembering (2.5), we derive the estimate (4.13). The estimate, together with the invariance of $\mathscr{A}_{s}$, completes the proof. □

Corollary 4.5. Under the assumptions of Theorem 4.4, the family of processes $\{U_{\sigma}(t, \tau)\}$, $\sigma\in\Sigma$ corresponding to Eq (1.1) has a compact uniform ($w.r.t. \sigma\in\Sigma$) attractor $\mathscr{A}_{s}^{\Sigma}$ which is bounded in the phase space $\mathscr{E}^{1}$.

Indeed, applying Theorems 4.3 and 4.4, we can state the result on the boundedness of the strong uniform attractor $\mathscr{A}_{s}^{\Sigma}$ in $\mathscr{E}^{1}$.

5.   Conclusions

We have investigated the dynamical behavior of a wave equation with time-dependent coefficients and quintic nonlinearity on a bounded domain, and established results on the existence and smoothness of a uniform attractor $\mathscr{A}_{s}^{\Sigma}$ in natural energy spaces $\mathscr{E}$.

There is still much work that needs to be done in this field. For example, the continuity of pullback attractors for Eq (1.1) with cubic nonlinearity $g$ was studied by Aragão et al. in ^[1]. Hence, a natural question is: Is it possible to otain similar results for Eq (1.1) when the non-linearity $g$ is assumed to have a sub-quintic or quintic rate? As we have already mentioned, the key difficulty in this problem is establishing the so-called ETS estimate (2.2). Nevertheless, we conjecture it is true, at least in the sub-quintic case. On the other hand, up to the moment, we do not know how to establish an ETS estimate in the quintic case, and this can be regarded as an open problem.

Author contributions

Feng Zhou: Conceptualisation, writing-original draft, formal analysis, project administration, writing-review and editing; Hongfang Li: Supervision and editiong; Kaixuan Zhu: Editing and project administration; Xin Li: Review and editing. All authors have read and approved the final version of the manuscript for publication.

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 authors would like to thank the anonymous referees for their interesting comments on this work. This work is supported by the NSFC of China (No. 11601522, 12201421); Natural Science Foundation of Shandong Province (ZR2021MA025, ZR2021MA028); the Hunan Province Natural Science Foundation of China (Grant Nos. 2022JJ30417, 2024JJ5288); the Provincial Natural Science Foundation of Hebei Grant No. A 2022203004; the Fund by Science Research Project of Hebei Education Department Grant No. QN2020203.

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.