Long-time dynamical behavior for a piezoelectric system with magnetic effect and nonlinear dampings

1.   Introduction

In this paper, we investigate the piezoelectric beam system with magnetic effect:

where $\varepsilon \in [0, 1]$, $\alpha = \alpha_{1}+\gamma^{2}\beta$, and $\alpha$, $\rho$, $\gamma$, $\beta$, $\mu > 0$ mean elastic stiffness, the mass density per unit volume, piezoelectric coefficient, water resistance coefficient, magnetic permeability, respectively, which is supplemented by the following initial boundary conditions:

It is well known that the ability to convert mechanical energy and electrical energy into each other is a characteristic of piezoelectric materials. In addition, it is smaller, less expensive, and more efficient ^[1]. Therefore, it has been applied in automotive ^[2,3], medical ^[4,5], space structures ^[6,7] and many other fields ^[8]. In the previous modeling stage, the magnetic effect was basically ignored due to its slight ^[9], but the latest results show that the magnetic effect may influence the performance of the system ^[10], so it is crucial to study piezoelectric beam models with magnetic effects. Nowadays, a growing number of scholars are beginning to do related research in this field.

The establishment of mathematical models of piezoelectric beams has a long history. In the early days, Tebou ^[11] and Haraux ^[12] established the model of a single beam due to Maxwell equation ^[13,14] and the dynamic interaction between electromagnetism is ignored. Therefore, they obtained the following equation:

Based on the magnetic effects and variational method, the equations of a single beam were derived by Morris and Özer ^[15], as follows:

where $V(t) = p_{t}(L, t)$ denotes electrical feedback controller. Then, they established the further results: the well-posedness of solution, strong stability of piezoelectric beam and so on ^[16,17].

In ^[18], Ramos et al. inserted a dissipative term $\delta v_{t}$ in the first equation of (1.3) and considered the following boundary condition

The authors applied energy method to prove the energy of the system and estimated that the energy is exponentially stable. After that, Ramos et al. changed the boundary condition in ^[19]:

and proved the exponential stability regardless of any relationship between system coefficients and there is equivalent to exact observability at the boundary.

Moreover, we know there are many ways to control piezoelectric vibration, among which time-delayed feedback control is a common way to improve stability of the system. Therefore, more and more researchers studied the influence of time-delayed effect on the stability of hyperbolic systems, and also proved the so-called destabilizing effect.

Dakto et al. ^{[20,21,22,23]} has obtained a series of research results in this direction. Among them, Dakto studied the equation as follows,

where $\tau > 0$, and proved that the time delay in the dissipative term charcatered by velocity term could make the beams lose stability.

Therefore, it is necessary to add some controlled term to stabilize the hyperbolic systems.

In 2021, Freitas and Ramos ^[24] considered the longitudinal vibration of the piezoelectric beams with thermal effect, magnetic effect and fraction damping:

the authors applied the semigroup theory to prove the well-posedness of solutions, then they showed that the existence of global attractor, exponential attractor and the upper semi-continuity of global attractor when $\nu\rightarrow0^{+}$.

Freitas et al. ^[25] studied nonlinear piezoelectric beam system with delay term:

and proved the system is asymptotically smooth gradient, and then they used stability estimation to obtain that the system is quasi-stable. Finally, they also proved the global attractor has finite fractal dimension.

Ma et al. ^[26] studied long-time dynamics of semilinear wave equation:

where $\Omega\subset\mathbb{R}^{3}$ is a bounded domain with a smooth boundary $\partial \Omega$ and $\varepsilon\in[0, 1]$. In this paper, they mainly proved the existence and properties of global attractors and attractors are continuous under autonomous perturbations. We also mention the pioneering work ^[27] that is related to nonlinear damping term in the research of attractor. In ^[27], Lasiecka and Ruzmaikina considered the existence, regularity and finite dimensionality of attractors for wave equations with some assumptions on nonlinear interior damping in 2-D case.

Freitas et al. ^[28] investigated the following wave equations:

where $\Omega\subset\mathbb{R}^{2}$ is a bounded domain with smooth boundary $\partial \Omega$, $\alpha_{1}, \alpha_{2}\in(0, 1)$ and $\varepsilon\in[0, 1]$. The authors showed that the dynamical system is quasi-stable and that the family of the global attractors is the continuous with respect to the parameter.

Recently, Freitas and Ramos ^[29] also considered a system modeling a mixture of three interacting continua with external forces with a parameter $\varepsilon\in[0, 1]$, and gave the smoothness of global attractors, the continuity of global attractors.

The main contributions of this paper are:

1) We consider the piezoelectric system with nonlinear damping and external forces with a parameter, we prove the existence of solutions by maximal monotone operator theory.

2) We establish a quasi-stability estimate to prove the system is quasi-stable and then we obtain the existence of exponential attractor and some properties of the global attractor.

3) We consider the upper semicontinuity of global attractors with respect to the parameter $\varepsilon\in[0, 1]$.

The present paper is organized as following. In Section 2, we mainly establish the well-posedness of the solution by nonlinear operator theory. In Section 3, we apply the infinite dynamical system theory to prove the existence of global attractors and exponential attractors. In Section 4, we mainly show the upper semi-continuity of the global attractors.

2.   Existence of global solutions

In this section, we first give some assumptions and notations. Then, we are concerned with well-posedness of the solution of the systems (1.1) and (1.2) by the similar argument as in ^[30,31,32] and the references therein.

2.1.   Assumption

Now, we give some assumptions which will be used hereinafter. In the following, we shall denote $C, C_{i}$ be the positive generic constants, which may be different in various lines. We assume that

(1) The function $F\in C^{2}(\mathbb{R}^2)$ satisfies

and there exists $\beta_{0}, m_{F}\geq 0$ such that

where

and $\beta_{1}$ is the embedding constant defined in (2.13). Moreover, there exist constant $r\geq1, C_{f} > 0$ such that

Furthermore, for arbitrary $v, p\in \mathbb{R}$, we get

(2) The external forces $h_{1}, h_{2}\in L^{2}(0, L)$, $\varepsilon\in[0, 1]$.

(3) For the nonlinear damping $g_{i}(i = 1, 2)$, we have

and there exists positive constants $m, M_{1}$ and $q\geq1$ such that

If $q\geq3$, there exist $l > q-1, M_{2} > 0$ such that

Furthermore, from (2.7), we obtain

Remark 2.1. A simple example of the function $F$ satisfying the assumption (1) can be

which implies that

It is easy to verify that

which yields that the assumptions $(2.2)$–$(2.4)$ hold with $m_F = \frac{1}{4}$ and $r = 4$. Noticing that

we can derive that $(2.5)$ also holds.

2.2.   The Cauchy problem

In the rest of this paper, we denote

We shall define the Sobolev Space

Because of $v(0) = 0$, we obtain the Poincaré's inequality

where $\lambda_{1} > 0$. Hence, we deduce an equivalent norm $\|v\|_{H^{1}_{*}(0, L)} = \|v_{x}\|_{2}$ in $H_*^1(0, L)$.

The energy space $\mathcal{H}$ is defined by

The inner product on $\mathcal{H}$ is

where $z = (v, p, v_{t}, p_{t})^{T}, \tilde{z} = (\tilde{v}, \tilde{p}, \tilde{v_{t}}, \tilde{p_{t}})^{T}$.

From the inner product, we can define the norm as

Moreover, there exists a constant $\kappa > 0$ such that

In fact, observing that

we have

where $\kappa = \max\{(2\gamma^{2}+1)\alpha^{-1}_{1}, 2\beta^{-1}\}$, the inequality (Eq 2.12) holds. Combining the Poincaré's inequality (Eq 2.10) and the above formula, there exists $\beta_{1} = \frac{\kappa}{\lambda_{1}} > 0$ such that

Let us write the systems (1.1) and (1.2) as an equivalent Cauchy problem

where $z_{0} = (v_{0}, p_{0}, v_{1}, p_{1})^{T}\in \mathcal{H},$ and $\mathcal{A}:D(\mathcal{A}) = \{(v, p, \tilde{v}, \tilde{p})^{T}\in \mathcal{H}: v, p\in H^{2}(0, L), \tilde{v}, \tilde{p}\in H^{1}_{*}(0, L), v_{x}(L) = p_{x}(L) = 0\}\subset\mathcal{H}\hookrightarrow\mathcal{H}$ is defined by

The function $\mathcal{F}:\mathcal{H}\rightarrow \mathcal{H}$ is defined by

By a simple calculation, we have

2.3.   Energy identities

Definition 2.1. A function $z(t) = (v, p, v_{t}, p_{t})^{T}\in C([0, \infty), \mathcal{H})$ with $z(0) = (v_0, p_0, v_1, p_1)^T$ is called a weak solution of the systems $(1.1)$ and $(1.2)$, if for any $\varphi, \psi\in H_{*}^{1}(0, L)$ it satisfies

Moreover, if

then $z$ is called the strong solution.

The total energy of solutions of (1.1) and (1.2) is defined by

where $E(t) = \frac{1}{2}\|z(t)\|_{\mathcal{H}}^{2}$.

Lemma 2.1. If $z = (v, p, v_{t}, p_{t})^{T}$ is a strong solution of $(1.1)$ and $(1.2)$, the following conclusions will hold,

(1)

(2) There exist constants $\beta_{2}, C_{F} > 0$ such that

Proof. Multiplying the equations in (1.1) by $v_{t}$, $p_{t}$, respectively, and using integration by parts, and applying the inequality (Eq 2.9), we can obtain (2.18).

Applying (2.2) and (2.14), we have

Using (2.3) and (2.20), we obtain

Let

since $\varepsilon\in[0, 1]$, we have

the first part of (2.19) is obtained with

Moreover, by (2.4) we have

By (2.13) and (2.25), we can deduce the second inequality in (2.19).

2.4.   Well-posedness

In this subsection, we will use the nonlinear operator theory to prove the well-posedness of solutions.

Definition 2.2. Let $X$ be a reflexive Banach space, the operator $A: X\rightarrow X'$ is called monotone if it satisfies

Furthermore, if $(A+I): X\rightarrow X'\ is\ onto$, then $A$ is maximal.

Definition 2.3. Let $X$ be a reflexive Banach space, the operator $B: X\rightarrow X'$ is called hemicontinuous, if

Lemma 2.2. The operator $\mathcal{F}: \mathcal{H}\rightarrow \mathcal{H}$ defined in $(2.15)$ is locally Lipschitz continuous.

Proof. Let the solutions $z^{1}, z^{2}\in \mathcal{H}$ such that $\|z^{1}\|_{\mathcal{H}}, \|z^{2}\|_{\mathcal{H}}\leq \mathcal{R}$, where $\mathcal{R} > 0$.

Then, we can deduce

It follows from (2.4) that

It follows from (2.24) that there exists some constant $C_{\mathcal{R}} > 0$ such that

which implies that

for some constant $C_{\mathcal{R}}' > 0$. This finishes the proof of this lemma.

Now, we are in the position to give the existence of the solutions.

Theorem 2.1. Suppose the assumptions $(2.1)$–$(2.7)$ hold. If $z_{0}\in \mathcal{H}$, the systems $(1.1)$ and $(1.2)$ have a unique weak solution $z(t)$ satisfies $z\in C([0, \infty); \mathcal{H})$, and it depends continuously on the initial data $z_{0}$. In addition, if $z_{0}\in D(\mathcal{A})$, the weak solution is a strong solution.

Proof. Similar to (2.15) and (2.16), using the monotonicity of function $g_{i}, i = 1, 2$, we know $\mathcal{A}$ is monotone. To obtain $\mathcal{A}$ is a maximal monotone operator, we need to show that there exists $z\in D(\mathcal{A})$, for arbitrary $w\in \mathcal{H}$ such that

In fact, in the following, we decompose the operator $\mathcal{A}$ as

where $B: D(B) = \{(v, p)\in H_{*}^{2}(0, L)\times H_{*}^{2}(0, L)\}\subset V\hookrightarrow H$,

$G: H\rightarrow H$,

Writing $z = (v, p, v_{t}, p_{t})^{T} = (\tau, \eta)^{T}\in V\times H$, $w = (w_{1}, w_{2})^{T}\in V\times H$, so

then, we can analyze that $\eta\in V$. Hence, we need to prove $K(\eta) = (B+I)\eta+G(\eta): V\rightarrow V'$ is onto. By Corollary 2.2 of ^[33], we need only to prove $K$ is maximal monotone and coercive. From (2.11) and the embedding $V\hookrightarrow H = H'\hookrightarrow V'$ we have

Firstly, let $\eta^{1}, \eta^{2}\in V$, then

By (2.9), we have

So we obtain that $B+I$ and $G$ are both monotone.

Secondly, let $\tau, \eta, \phi\in V$, $\lambda\in \mathbb{R}$, we have

which implies that the continuity of $\langle (B+I)(\tau+\lambda \eta), \phi\rangle$ at $\lambda = 0$. Moreover, we have

Clearly

From (2.7) and Dominated Convergence Theorem, we deduce

So we obtain that $B+I$ and $G$ are hemicontinuous.

In addition, we can deduce $B+I$ and $G$ are coercive from (2.25) and (2.26).

Therefore, according to Theorem 2.6 of ^[33], we know that $K$ is coercive and maximal monotone, which implies that $K$ is onto. That is, $\mathcal{A}$ is maximal monotone in $\mathcal{H}$.

In conclusion, since $\mathcal{A}$ is maximal monotone and $\mathcal{F}$ is locally Lipschitz, by applying Theorem 7.2 of ^[34], we can obtain: when $z_{0}\in \mathcal{H}$, the problem (2.14) has a unique weak solution $z(t)\in C([0, t_{max}); \mathcal{H})$. Moreover, if $t_{max} < \infty$, then $\lim\sup_{t\rightarrow t_{max}^{-}}\|z\|_{\mathcal{H}} = \infty$. When $z_{0}\in D(\mathcal{A})$, the problem (2.14) has a unique strong solution $z(t)\in C([0, t_{max}), D(\mathcal{A})), \ (t_{max}\leq\infty)$.

Now, we need to show the existence of global solutions, that is $t_{\max} = \infty.$ In fact, let $z(t)$ be a strong solution defined in $[0, t_{\max})$. From (2.18), we infer

It follows from (2.19) and (2.17) that

By density argument, the conclusion also holds for weak solution. Therefore $t_{max} = \infty$.

Finally, let $z^{1}, z^{2}$ be two weak solutions, by the standard arguments, for any $T > 0$, there exists constant $C > 0$ such that

The proof is complete.

3.   The existence of global attractor

In this section, for the sake of completeness, we collect some known results in the theory of nonlinear dynamical systems (see ^{[35,36,37,38]}).

Let $(v, p, v_{t}, p_{t})^{T}$ be the unique solution for the systems (1.1) and (1.2). We can define the operator $S(t):\mathcal{H}\rightarrow \mathcal{H}$ by

Hence, $(\mathcal{H}, S(t))$ constitutes a dynamical system.

Definition 3.1. Let $B\subset \mathcal{H}$ be a positively invariant set of a dynamical system $(\mathcal{H}, S(t))$.

1. A function $\Phi(\mathfrak{y})$ is said to be a Lyapunov function, if $t\rightarrow \Phi(S(t)\mathfrak{y})$ is a non-increasing function for any $\mathfrak{y}\in B$.

2. A Lyapunov function is called strict, if there exist $t_{0} > 0$, $\mathfrak{y}\in B$ such that $\Phi(S(t_{0})\mathfrak{y}) = \Phi (\mathfrak{y})$, then $\mathfrak{y} = S(t)\mathfrak{y}, \ t > 0$.

Definition 3.2. The dynamical system $(\mathcal{H}, S_{t})$ is quasi-stable on $B\subset\mathcal{H}$, if there exist a compact seminorm $\chi_{X}(\cdot)$ on the space $X$ and nonnegative scalar function $\mathfrak{a}(t), \mathfrak{b}(t), \mathfrak{c}(t)\in\mathbb{R^{+}}$ satisfy:

(1) $\mathfrak{a}(t), \mathfrak{c}(t)$ are locally bounded on $[0, \infty)$;

(2) $\mathfrak{b}(t)\in L^{1}(\mathbb{R^{+}})$, and $\lim_{t\rightarrow \infty}\mathfrak{b}(t) = 0$;

(3) for any $\mathfrak{y}_{1}, \mathfrak{y}_{2}\in B$ and $t\geq0$, the estimates

and

hold, where $S(t) = (\mathfrak{u}^{i}(t), \mathfrak{u}_{t}^{i}(t)), i = 1, 2$.

Lemma 3.1. The dynamical system $(\mathcal{H}, S(t))$ is gradient, that is, there exists a strict Lyapunov function $\Phi\in \mathcal{H}$. What's more,

Proof. Let $\mathcal{E}(t)$ defined in (2.17) be a Lyapunov function $\Phi$ and $z_{0} = (v_{0}, p_{0}, v_{1}, p_{1})^{T}\in\mathcal{H}$, we can infer that $t\rightarrow \Phi(S(t)z_{0})$ is a non-increasing function from (2.18).

Supposing that $\Phi(S(t)z_{0}) = \Phi(z_{0})$, for $t\geq0$, we have

We obtain

Consequently, $v_{t}(t) = v_{0}, p_{t}(t) = p_{0}$, which implies $S(t)z_{0} = z(t) = (v_{0}, p_{0}, 0, 0)^{T}$ is a stationary solution of $(\mathcal{H}, S(t))$.

From (2.19), we obtain

Let $\Phi(z)\rightarrow \infty$, we have $\|z\|_{\mathcal{H}}\rightarrow \infty$. Then, it follows from (2.19) that

As a result, we infer from $\|z\|_{\mathcal{H}}\rightarrow \infty$ that $\Phi(z)\rightarrow \infty$.

Lemma 3.2. The set of stationary points $\mathcal{N}$ of $S(t)$ is bounded in $\mathcal{H}$.

Proof. Let $z = (v, p, 0, 0)^{T}\in \mathcal{N}$ be the stationary solution of systems (1.1) and (1.2). Then, we have the following elliptic equations

Multiplying the equations in (3.1) by $v$, $p$, respectively, and integrating the result over $(0, L)$, we have

Hence, using (2.1), (2.2), and (2.5), we obtain

By (2.21), we have

By Young's inequalities and (2.13), we infer

Therefore, we conclude

The proof is complete.

Lemma 3.3. Suppose that the Assumption 2.1 holds. Let $B$ be a bounded forward invariant set in $\mathcal{H}$ and $S(t)z^{i} = (v^{i}, p^{i}, v_{t}^{i}, p_{t}^{i})^{T}$ be a weak solution of the systems $(1.1)$ and $(1.2)$ with $z_{0}^{i}\in B, i = 1, 2$. Then there exist constant $\sigma, \varsigma, C_{B} > 0$ independent of $\varepsilon$ such that

where $E(t) = \frac{1}{2}\|z\|_{\mathcal{H}}^{2}$, $v = v^{1}-v^{2}, p = p^{1}-p^{2}, \theta\geq2$.

Proof. Let $F_{i}(v, p) = f_{i}(v^{1}, p^{1})-f_{i}(v^{2}, p^{2}), i = 1, 2, G_{1}(v_{t}) = g_{1}(v_{t}^{1})-g_{1}(v_{t}^{2}), G_{2}(p_{t}) = g_{2}(p_{t}^{1})-g_{2}(p_{t}^{2})$. Then $v = v^{1}-v^{2}, p = p^{1}-p^{2}$ satisfy

Multiplying the first equation of (3.3) by $v$, the second one by $p$, and integrating over $[0, L]\times [0, T]$, we have

Step 1. By Poincáre's and Hölder's inequalities, we have

Then

where ${C}$ is a positive constant. From (2.9), we conclude

Step 2. According to Young's inequality and (2.9), we can deduce

Applying (2.7), we infer

For further estimation, we divide it into three cases.

Case a: $q = 1$. It is easy to get

Case b: $1 < q < 3$. Applying the assumption (2.9) and Hölder's inequality, we have

where $d = \frac{2}{q-1}$, and $\frac{1}{d}+\frac{1}{d_{1}} = 1$.

Case c: $q\geq3$. Using the assumption (2.8), by the similar argument as in Case b, let $d = \frac{l}{q-1} > 1$, we can obtain the same result.

Combining the estimates of the above three cases, we can deduce that there exist ${C} > 0, \theta\geq2$ such that

Similarly,

Furthermore, due to $z^{1}, z^{2}\in B$, and by (2.18), (2.19), we conclude there exists ${C}_{B} > 0$ such that

Combining with (2.8), (3.4), (3.5), and applying $L^{2\theta}(0, L)\hookrightarrow L^{\theta}(0, L)$, we deduce

Step 3. Applying (2.4), $H_{*}^{1}(0, L)\hookrightarrow L^{r}(0, L), r\in(1, \infty)$ and Hölder's inequality, we can infer

Similarly, we obtain

Combining (3.7) with (3.6), there exists ${C}_{T} > 0$, we have

Therefore, combining the above estimates, we have

for some constants ${C}_{B}, {C}_{B, T} > 0$.

Step 4. Multiplying the equations of (3.3) by $v_{t}, p_{t}$, respectively, and integrating over $[0, L]\times[s, T]$, we obtain

Due to

we obtain

By the similar argument, we have

Analogously,

Consequently,

Let $\zeta = \frac{1}{T}$, we have

Then, integrating in $[0, T]$, there exists a constant ${C}_{B, T} > 0$ such that

Step 5. Let (3.9) with $s = 0$ and (3.10) with $\zeta = 1$, we have

Combining the above estimates with (3.8), we can deduce

Then, from (3.11)

Choosing $T > 2{C}_{B}$, letting $m_{T} = \frac{{C}_{B}}{T-{C}_{B}} < 1$, we obtain that

Let $M_{n} = \sup_{s\in[nT, (n+1)T]}(\|v(s)\|^{2}_{2\theta}+\|p(s)\|^{2}_{2\theta}), n\in\mathbb{N}$. Then, repeating the above argument progress on any $[nT, (n+1)T]$, we have

For any $t\geq0$, there exists $n\in\mathbb{N}, k\in[0, T)$ such that $t = nT+k$. Hence, we have

Consequently, letting $\sigma = -\frac{ln(m_{T})}{T}, \varsigma = m_{T}^{-1}, C_{B} = \frac{{C}_{B, T}}{1-m_{T}}$, we have

The proof is complete.

Lemma 3.4. Let $B\subset\mathcal{H}$ be a bounded positively invariant set, then dynamical system $(\mathcal{H}, S(t))$ is quasi-stable.

Proof. Defining $S(t)z^{i} = (v^{i}(t), p^{i}(t), v_{t}^{i}(t), p_{t}^{i}(t))^{T}$ for $z^{i}\in B, i = 1, 2$, and $v = v^{1}-v^{2}, p = p^{1}-p^{2}$. Then, it follows from (2.28) that

where $\mathfrak{a}(t) = e^{CT}$.

Now, we let $X = H_{*}^{1}(0, L)\times H_{*}^{1}(0, L)$ and define the semi-norm

Since $H_{*}^{1}(\Omega)\hookrightarrow\hookrightarrow L^{2\theta}(\Omega)$, we can obtain that $\chi_{X}$ is a compact semi-norm on $X$.

According to Lemma 3.3, we have

with $\mathfrak{b}(t) = \varsigma e^{-\sigma t}$, $\mathfrak{c}(t) = C_{B}$.

It is easy to verify that $\mathfrak{b}(t)\in L^{1}(\mathbb{R}^{+})$ and $\lim_{t\rightarrow \infty}{\mathfrak{b}(t)} = 0$.

Then, since $B$ is a bounded subset of $\mathcal{H}$, we have $\mathfrak{c}(t)$ is locally bounded on $[0, \infty)$. By the Definition 3.2, we have the dynamical system is quasi-stable on $B\subset\mathcal{H}$.

Since dynamical system $(\mathcal{H}, S(t))$ is quasi-stable, we can give our main results as following.

Theorem 3.1. Under the assumptions of Theorem 2.1, we obtain

(1) The dynamical system $(\mathcal{H}, S(t))$ has a global attractor $\mathcal{A} \subset \mathcal{H}$ which is compact and connected. Moreover, the attractor can be characterized by the unstable manifold

emanating from the set of stationary solutions established in Lemma 3.2.

(2) The attractor $\mathcal{A}$ has finite fractal dimension $dim_{\mathcal{H}}\mathcal{A}$.

(3) Every trajectory stabilizes to the set $\mathcal{N}$, that is,

In particular, there exists a global minimal attractor $\mathcal{A}_{min}$ to the dynamical system, whichis precisely characterized by the set of the stationary points $\mathcal{N}$, that is $\mathcal{A}_{min} = \mathcal{N}$.

(4) The attractor is bounded in $\mathcal{H}_{1} = (H^{2}(0, L)\cap H_{*}^{1}(0, L))^{2}\times (H_{*}^{1}(0, L))^{2}$, and every trajectory $z = (v, p, v_{t}, p_{t})^{T}$ in $\mathcal{A}$ has the property

for some constant $R > 0$ independent of $\varepsilon\in[0, 1]$.

Proof. (1) It follows from Lemma 3.4 that $(\mathcal{H}, S(t))$ is quasi-stable. Hence, we have that $(\mathcal{H}, S(t))$ is asymptotically smooth by Proposition 7.9.4 of ^[36]. Then, applying Lemmas 3.1, 3.2 and Corollary 7.5.7 of ^[36], we can conclude $(\mathcal{H}, S(t))$ possesses a compact global attractor $\mathcal{A}$. In addition, it can be characterized by $\mathcal{A} = \mathbb{M}^{+}(\mathcal{N}).$

(2) Since the system $(\mathcal{H}, S(t))$ is quasi-stable, applying Theorem 7.9.6 of ^[36], we conclude that attractor $\mathcal{A}$ has finite fractal dimension $dim_{\mathcal{H}}\mathcal{A}$.

(3) Combining Theorem 3.1(1) and Theorem 7.5.10 of ^[36], we can get the desired result immediately.

(4) Since $(\mathcal{H}, S(t))$ is quasi-stable on $\mathcal{A}$, the arbitrary trajectory $z = (v, p, v_{t}, p_{t})^{T}$ in $\mathcal{A}$ has the following regularity properties

It follows from (1.1) that

Then, we can deduce

Using the fact $\alpha_{1}\neq0$, $v_{t}, p_{t}\in L^{\infty}(\mathbb{R}; H_{*}^{1}(0, L))\hookrightarrow L^{\infty}(\mathbb{R}; L^{2}(0, L))$ and $f_{i}(v, p)$ is locally Lipschitz continuous, we have

It follows from (3.13) that

The proof is complete.

Theorem 3.2. The system $(\mathcal{H}, S(t))$ has a generalized exponential attractor. More precisely, for any given $\xi\in(0, 1]$, thereexists a generalized exponential attractor $\mathcal{A}_{exp, \xi} \subset \mathcal{H}$ with finite fractal dimension in the extended space $\mathcal{H}_{-\xi}$ which is defined as the interpolation of

Proof. Let us take $\mathcal{B} = \{z\in \mathcal{H}|\Phi(z)\leq R\}$ where $\Phi$ is the strict Lyapunov functional given in Lemma 3.1. Then we can derive that for sufficiently large $R$ that $\mathcal{B}$ is a positively invariant bounded absorbing set, which shows that the system is quasi-stable on the set $\mathcal{B}$.

Then for solution $z(t) = S(t)z_0$ with initial data $z(0)\in \mathcal{B}$, we can derive that, for any $T > 0$,

which shows that

where $\mathcal{C}_{\mathcal{B}T}$ is a positive constant and $t_{1}, t_{2}\in[0, T]$. Hence we obtain that for any initial data $z_0\in \mathcal{H}$ the map $t\mapsto S(t)z_0$ is $\frac{1}{2}$-Hölder continuous in the extended phase space $\mathcal{H}_{-1}$. Therefore, it follows from Theorem 7.9.9 in ^[34] that the dynamical system $(\mathcal{H}, S(t))$ possesses a generalized fractal exponential attractor with finite fractal dimension in the extended space $\mathcal{H}_{-1}$.

Furthermore, by the standard interpolation theorem, we can obtain the existence of exponential attractors in the extended space $\mathcal{H}_{-\xi}$ with $\xi\in (0, 1)$. The proof is complete.

4.   Upper semicontinuity of global attractor

In this section, we denote the attractor obtained in Theorem 3.1 as the $\mathcal{A}_{\varepsilon}$. Then, we investigate the upper semicontinuity of the attractors $\mathcal{A}_\varepsilon$ as $\varepsilon\rightarrow \varepsilon_0$.

Definition 4.1. ^[26] Let $\Lambda$ be a complete metric space and $S_{\sigma}(t)$ a family of semigroups on $\mathcal{H}$, where $\sigma\in\Lambda$. The global attractors $\mathcal{A}_{\sigma}$ is called upper semicontinuous on $\sigma_{0}\in\Lambda$ if

where $dist_{\mathcal{H}}\{\mathbb{A}, \mathbb{B}\} = \sup_{a\in \mathbb{A}}\inf_{b\in \mathbb{B}}d(\mathbb{A}, \mathbb{B})$ expresses the Hausdorff semi-distance in $X$. Similarly, $\mathcal{A}_{\sigma}$ is lower semicontinuous on $\sigma_{0}\in\Lambda$ if

Then $\mathcal{A}_{\sigma}$ is continuous on $\sigma_{0}\in\Lambda$ if

where $d_{\mathcal{H}}(\mathbb{A}, \mathbb{B}) = \max\{dist_{\mathcal{H}}(\mathbb{A}, \mathbb{B}), dist_{\mathcal{H}}(\mathbb{B}, \mathbb{A})\}$ expresses the Hausdorff metric in $X$.

Proposition 4.1. ^[39] Assume that

(H1) $(\mathcal{H}, S_{\sigma}(t))$ has a global attractor $\mathcal{A}_{\sigma}$ for any $\sigma\in\Lambda$,

(H2) There exists a bounded set $B\subset\mathcal{H}$ such that $\mathcal{A}_{\sigma}\subset B$ for every $\sigma\in\Lambda$,

(H3) $S_{\sigma}(t)z$ is continuous in $\sigma$ for $t > 0$ and uniformly for $z$ in bounded subsets of $\mathcal{H}$.

Then the global attractor is continuous on all $I$, where $I$ is a "residual" set dense in $\Lambda$.

Lemma 4.1. There exists a set $I$ dense in $[0, 1]$ such that the global attractor $\mathcal{A}_{\varepsilon}$ obtained in Theorem 3.1 is continuous at $\varepsilon_{0}\in I$, that is

Proof. The argument is inspired by ^[40,41]. We apply Proposition 4.1 with $\Lambda = [0, 1]$. Then Theorem 3.1 implies that the assumption (H1) holds.

It follows from (2.19) and the fact $\sup_{z\in\mathcal{A}_{\varepsilon}}{\Phi_{\varepsilon}(z)}\leq \sup_{z\in\mathcal{N}_{\varepsilon}}\Phi_{\varepsilon}(z)$ that

Hence, we can derive from (3.2) that there exists a positive constant $C$ independent of $\varepsilon$ such that

Then we have that $\mathcal{B} = \{z\in\mathcal{H}:\|z\|_{\mathcal{H}}^{2}\leq C\}$ is a bounded set which is independent of $\varepsilon$ and $\mathcal{A}_{\varepsilon}\subset \mathcal{B}$ for any $\varepsilon\in [0, 1]$. Therefore, the assumption (H2) holds.

Let $B$ be a bounded set of $\mathcal{H}$. Then for any given $\varepsilon_{1}, \varepsilon_{2}\in [0, 1], z_{0}\in B$, we define

and

Then $(v, p, v_t, p_t)$ satisfies the following equations

where $F_i(v, p)$ ($i = 1, 2$) $G_1(v_t)$ and $G_2(p_t)$ are constructed by the same as in Lemma 3.3. Multiplying the equations of (4.2) by $v_{t}, p_{t}$, respectively. and integrating over $[0, L]$ by parts, we have

Using (2.4), Hölder's inequality and $H_{*}^{1}(0, L)\hookrightarrow L^{r}(0, L), r\in[0, \infty)$, we can derive that

Since $\mathcal{E}_{\varepsilon}(t)$ is a non-increasing function, then for any $z_{0}\in B$, we have

Combining the above estimate with (4.4) and (2.12), we have

Similarly,

Therefore,

It follows from the monotonicity property (2.9) that

Moreover, it is easy to verify that

Combining (4.5)–(4.7) with (4.3), we have

Using the Gronwall's inequality and the fact $\|z(0)\|_{\mathcal{H}}^{2} = 0$, we can derive from (4.8) that

Hence we have

So the assumption (H3) holds. As a conclusion, we obtain that there exists a dense set $I\subset [0, 1]$ such that (4.1) holds by Proposition 4.1.

Theorem 4.1. Suppose the assumptions of Theorem 3.1 hold. Then the family of global attractors $\mathcal{A}_{\varepsilon}$ is upper semicontinuous at $\varepsilon_0$, namely,

Proof. The argument is inspired by ^[42,43]. Firstly, we suppose that (4.9) does not hold. Then there exist $\lambda > 0$, the sequence $\varepsilon_{n}\rightarrow \varepsilon_{0}$ and $z_{0}^{n}\in \mathcal{A}_{\varepsilon^{n}}$ such that

Let $z^{n}(t) = (v^{n}(t), p^{n}(t), v_{t}^{n}(t), p_{t}^{n}(t))^{T}$ be a bounded full trajectory from the attracator $\mathcal{A}_{\varepsilon^{n}}$ with $z^{n}(0) = z_{0}^{n}$. It follows from (3.12) that ${z^{n}}$ is bounded in $L^{\infty}(\mathbb{R}; \mathcal{H}_{1})$.

Noticing $\mathcal{H}_{1}\hookrightarrow\hookrightarrow\mathcal{H}$, applying Simon's Compactness Theorem (see ^[44] for details), we can obtain that there exist $z\in C([-T, T]; \mathcal{H})$ and a subsequence (still denote)$\{z^{n}\}$ such that,

Then, we can conclude that

Let $z(t) = (v(t), p(t), v_{t}(t), p_{t}(t))^{T}$ be a bounded full trajectory of the limiting semi-flow. Then, we can infer that $z$ solves the limiting equation ($\varepsilon = \varepsilon_{0}$), namely,

In fact, from (1.1), we can get $z^{n}$ satisfies

we can use the same argument as in the proof of (H3) in Lemma 4.1, so we can infer that (4.11) is the limit case of (4.12) as $n\rightarrow \infty$.

Consequently, we have

which contradicts (4.10). This completes the proof.

Acknowledgments

The authors would like to thank the referees for the careful reading of this paper. This project is supported by NSFC (No.11801145 and No.12101189), the Innovative Funds Plan of Henan University of Technology 2020ZKCJ09 and the Fund of Young Backbone Teacher in Henan Province (No.2018GGJS068).

Conflict of interest

The authors declare there is no conflicts of interest.