Long-time behavior of a class of viscoelastic plate equations

1.   Introduction

In this paper, we study the following initial-boundary value problem for nonlinear viscoelastic plate equations

where $\Omega$ is a bounded domain of ${\mathbb R}^N$ with a smooth boundary $\partial \Omega$, the memory kernel $g$ and the external forces $f$, $h$ will be specified later.

Problem 1-3 can be used to describe the vibrations of viscoelastic materials possessing a capacity of storage and dissipation of mechanical energy, see [17] for the details. And $u(x,t)$ represents the displacement at time $t$ of a particle having position $x$ in a given reference configuration with the prescribed past history $u_0:\Omega\times(-\infty,0]\rightarrow \mathbb{R}$. In view of the main results of [5], we see that the viscoelastic term (namely the memory term) produces the effect of strong dissipation, which prevails the effect of weak damping term on the decay of solutions in time.

There have been many works on the long-time behavior of viscoelastic plate equations, we refer the readers to [1,2,4,6,18,20,21,25] and the references therein. As for viscoelastic plate equations with past history, Pata [23] studied

where $A$ is a self-adjoint and strictly positive linear operator, $\alpha$ and $\mu$ are positive constants. Based on certain assumptions on $g$, he analyzed the exponential stability of the related semigroup. Guesmia and Messaoudi [13] investigated

Under some assumptions on $A$ and $g$, they established a general decay result which depends on the behavior of $g$. Jorge Silva and Ma [15] considered

where $h\in L^2(\Omega)$, and $\Omega$ is a bounded domain of ${\mathbb R}^N$ ($N\geq1$) with a smooth boundary $\partial \Omega$. Under some assumptions on $f$ and $g$, they obtained the global well-posedness and regularity of weak solutions. Moreover, they proved the exponential decay of energy. Recently, Conti and Geredeli [9] studied

where $h\in L^2(\Omega)$, $\Omega$ is a bounded domain of ${\mathbb R}^3$ with a smooth boundary $\partial \Omega$. Under some assumptions on $f_1$, $f_2$ and $g$, they obtained the existence and regularity of global attractors.

In the works mentioned above, authors introduced a variable which reflects the relative displacement history so that the corresponding problem could be turned into an autonomous system. This scheme is so-called the past history approach [12] which suggests to consider some past history variables as additional components of the phase space corresponding to the equation under study.

In the present paper, in order to study the long-time behaviour of solutions of problem 1-3, we employ the past history approach and the operator technique so that Eq. 1 can be transformed into an abstract system in the history phase space. And thus the operator technique combined with the energy estimates becomes a crucial tool for the proof of the existence of global attractors.

This paper is organized as follows. In Section 2 some notations and assumptions on $f$ and $g$ are displayed. Moreover, 1-3 is transformed into a generalized problem, and the main results of this paper are stated. In Section 3 the global well-posedness of regular solutions is obtained. And the global well-posedness of weak solutions is established by the density arguments [6]. In Section 4 the existence of global attractors is derived by means of the existence of an absorbing set and the semigroup decomposition [14,16,26].

2.   Preliminaries and main results

2.1.   Notations and assumptions

Throughout the paper, in order to simplify the notations, we denote

$(\cdot,\cdot)$ denotes either the $L^2$-inner product or a duality pairing between a space and its dual space. Moreover, $|\Omega|$ stands for the Lebesgue measure of $\Omega$, $C_i$, $i = 1,2,3,\cdots$ denote some different positive constants, and $\mathfrak{C}_i$, $i = 1,2,3$ represent the positive constants for inequalities

As in [3,10,27], we give the following assumptions on $f$ in order to state the main results of this paper.

$(\mathbb{A}_1)$: $f(0) = 0$, and there exists a constant $b>0$ such that

where

Moreover,

and

where $0<\varrho<1$ and

In addition, as in [7,19,22], we assume that $g$ satisfies the following conditions.

$(\mathbb{A}_2)$: $g\in C^1(\mathbb{R}^+)\cap L^1(\mathbb{R}^+)$, $g(t)\geq0$, $g^\prime(t)\leq 0$, $t\in[0,\infty)$, and

2.2.   Reformulation of the problem

As in [2,3,8,10,27], we define the operator $A:D(A)\subset L^2(\Omega)\rightarrow L^2(\Omega)$

where the dense domain

It is easy to verify that $A$ is self-adjoint and strictly positive. Thus $A^\gamma$ is also self-adjoint and strictly positive for any $\gamma>0$. Denote $V_{4\gamma}: = D(A^\gamma)$ and $V_{-\gamma}: = V_\gamma^\prime$. Then, for any $\gamma\in \mathbb{R}$, $V_\gamma$ and $\mathcal{L}_{g,\gamma}$ are Hilbert spaces equipped with inner products and norms

where

Thus

In this way, problem 1-3 can be seen as

Now we are in a position to define the auxiliary function

Thus the viscoelastic dissipation in 7 can be rewritten as

Therefore, problem 1-3 is transformed into the following system

with

where

Definition 2.1. $(u(t),u_t(t),v^t)$ is called a weak solution of problem 8, 9 if $u\in C([0,T];V_2)$, $u_t\in C([0,T];V_0)$, $v^t\in C([0,T];\mathcal{L}_{g,2})$, $u(0) = u_0$ in $V_2$, $u_t(0) = u_1$ in $V_0$, $v^0 = v_0$ in $\mathcal{L}_{g,2}$, and

for any $w_1\in V_2$, $w_2\in \mathcal{L}_{g,2}$ and a.e. $t\in(0,T]$. }

Thus, in order to deal with problem 1-3, we study the modified problem 8, 9. In fact, for a solution $(u,u_t,v^t)$ of problem 8, 9, we have

In view of [17,Chapter 2,Section 4], we see that $g(t): = -G^\prime(t)$, where $G(t)$ is the viscoelastic flexural rigidity. From

it follows that

Since

and

we deduce that

Substituting this into 10, we obtain

Due to

we conclude that

which shows that $(u,u_t)$ is a solution of problem 1-3.

2.3.   Statement of main results

The main results of this paper are stated as follows.

Theorem 2.2. Let $\rm (A_1)$ and $\rm (A_2)$ be fulfilled. Assume that $h\in V_0$ and $(u_0,u_1,v_0)\in Z: = V_2\times V_0\times \mathcal{L}_{g,2}$. Then problem 8, 9 admits a unique solution $(u,u_t,v^t)\in C([0,\infty);Z)$ depending continuously on initial data.

Define the mapping $S(t):Z\rightarrow Z$ by

Then it is easy to see from Theorem 2.2 that $\{S(t)\}_{t\geq0}$ is a $C^0$-semigroup generated by problem 8, 9.

Theorem 2.3. Let $\rm (A_1)$ and $\rm (A_2)$ be fulfilled. And there exists a constant $\rho>0$ such that $g^\prime(t)+\rho g(t)\leq 0$ for all $t\in[0,\infty)$. Assume that $h\in V_0$ and $(u_0,u_1,v_0)\in Z$. Then $S(t)$ possesses a global attractor in $Z$.

3.   Proof of Theorem 2.2

Theorem 3.1. Let $\rm (A_1)$ and $\rm (A_2)$ be fulfilled. Assume that $h\in V_0$, $u_0\in V_3$, $u_1\in V_1$, $v_0\in \mathcal{L}_{g,3}$. Then problem 8, 9 admits a unique solution $u\in L^\infty(0,\infty;V_3)$, $u_t\in L^\infty(0,\infty;V_1)\cap L^2(0,\infty;V_2)$, $v^t\in L^\infty(0,\infty;\mathcal{L}_{g,3})$, which depends continuously on initial data.

Proof. Let $\{\omega_j\}_{j = 1}^\infty$ be an orthogonal basis of $V_2$ and an orthonormal basis of $V_0$ given by eigenfunctions of $A$. As in [11,15], we select $\{e_j\}_{j = 1}^\infty$ of the form $\{l_k\tilde{\omega}_j\}_{k,j = 1}^\infty$, where $\{l_k\}_{k = 1}^\infty$ is an orthonormal basis of $L_g^2(\mathbb{R}^+)\cap C_0^\infty(\mathbb{R}^+)$ and $\tilde{\omega}_j = \frac{\omega_j}{\|\omega_j\|_{V_2}}$. Then $\{e_j\}_{j = 1}^\infty$ is an orthonormal basis of $\mathcal{L}_{g,2}$.

We construct the approximate solutions of problem 8, 9

which satisfy

with

The approximate problem 11, 12 can be reduced to an ordinary differential system in the variables $\xi_{jn}(t)$ and $\zeta_{jn}(t)$. In terms of standard theory for ODEs, there exists a solution $(u_n(t),u_{nt}(t),v^t_n)$ on some interval $[0,T_n)$ with $T_n\leq T$. The following estimates will allow us to extend the local solutions to $[0,T]$ with any $T>0$.

Replacing $\omega_j$ in 11$_1$ with $u_{nt}$ and $e_j$ in 11$_2$ with $v^t_n$, summing for $j$ and adding the two results, we obtain

where

Since $\lim\limits_{\tau\rightarrow 0} v^t_n(\tau) = 0$, we deduce from $\rm (A_2)$ that

Hence, by integrating 13 with respect to $t$ from $0$ to $t$, we get

It follows from 4 in $\rm (A_1)$ that, for any $\eta>0$, there exists a constant $C_\eta>0$ such that

By virtue of Cauchy's inequality with $\epsilon>0$, we get

Consequently, taking sufficiently small $\eta$ and $\epsilon$ such that

we deduce from 14 that

Hence, from 15, 16 and 12, it follows that

Replacing $\omega_j$ in 11$_1$ with $A^{\frac{1}{2}} u_{nt}$ and $e_j$ in 11$_2$ with $A^{\frac{1}{2}} v^t_n$, summing for $j$ and adding the two results, we obtain

Noting that

and

we conclude from 17 that

Therefore, there exist $u$, $v^t$ and subsequences of $\{u_n\}$, $\{v^t_n\}$, still represented by the same notations and we shall not repeat, such that, as $n\to\infty$,

for any $T>0$. According to the Aubin-Lions lemma, we have

Moreover, from 18-20, it follows that

We now claim that for any $t\in [0,T]$ and fixed $j$,

as $n\rightarrow\infty$.

Indeed, for any $w\in V_2$, we have

If $p>2$, then when $N\leq 4$,

when $N>4$,

Hence

If $p = 2$, then it is clear that 23 remains valid. Therefore,

Thus assertion 22 follows from 21.

For fixed $j$,

Hence

Consequently, for fixed $j$, integrating 11 with respect to $t$ and taking $n\rightarrow \infty$, we get

Moreover, it is easy to see from 12 that $u(0) = u_0$ in $V_3$, $u_t(0) = u_1$ in $V_1$, $v^0 = v_0$ in $\mathcal{L}_{g,3}$. Therefore, $(u,u_t,v^t)$ is a solution of problem 8, 9.

Next we prove continuous dependence of $(u(t),u_t(t),v^t)$ on $(u_0,u_1,v_0)$. Suppose that $(u,u_t,v^t)$ and $(\bar{u},\bar{u}_t,\bar{v}^t)$ are two regular solutions of problem 8, 9 with initial data $(u_0,u_1,v_0)$ and $(\bar{u}_0,\bar{u}_1,\bar{v}_0)$, respectively. Set $\tilde{u} = \bar{u}-u$ and $\tilde{v}^t = \bar{v}^t-v^t$. Then

By the arguments similar to [7,Lemma 4.9], we obtain

By the arguments similar to the proof of 23, we have

Note that $-(\tilde{v}^t_\tau,\tilde{v}^t)_{g,2}\leq0$. Hence, by taking $\epsilon = \frac{1}{C_9}$, we deduce from 25 that

As a consequence, by Gronwall's inequality, we obtain

In particular, by taking $(u_0,u_1,v_0) = (\bar{u}_0,\bar{u}_1,\bar{v}_0)$, it is obvious that $(u,u_t,v^t)$ is the unique solution of problem 8, 9.

Proof of Theorem 2.2. For $u_0\in V_2$, $u_1\in V_0$, $v_0\in \mathcal{L}_{g,2}$, there exist $\{u_{0m}\}\subset V_3$, $\{u_{1m}\}\subset V_1$, $\{v_{0m}\}\subset \mathcal{L}_{g,3}$ such that

According to Theorem 3.1, for any $m\in\mathbb{N^+}$, problem 8, 9 admits a unique regular solution $(u_m,u_{mt},v^t_m)$ satisfying

Hence $u_m\in C([0,T];V_2)$, $u_{mt}\in C([0,T];V_0)$. Moreover, according to [24,Theorem 3.2], we have $v_m^t\in C([0,T];\mathcal{L}_{g,2})$.

Set $y_m = u_{m_2}-u_{m_1}$ and $z_m = v^t_{m_2}-v^t_{m_1}$. Then, by the arguments similar to the proof of 26, we get

By 27 and the arguments similar to the proof of 17, we have

Thus $(u,u_t,v^t)$ is a global weak solution of problem 8, 9.

Suppose that $(u,u_t,v^t)$ and $(\bar{u},\bar{u}_t,\bar{v}^t)$ are two solutions of problem 8, 9 with initial data $(u_0,u_1,v_0)$, $(\bar{u}_0,\bar{u}_1,\bar{v}_0)$, respectively. Then there exist

such that

Set $\tilde{u}_m = \bar{u}_m-u_m$ and $\tilde{v}_m^t = \bar{v}_m^t-v_m^t$. Then, on account of 26, we obtain

Therefore, in terms of 28-32, the conclusions of Theorem 2.2 are derived immediately.

4.   Proof of Theorem 2.3

In this section, for the sake of convenience, we denote

Lemma 4.1. Under the conditions of Theorem 2.3, $S(t)$ possesses an absorbing set in $Z$.

Proof. Let $U = u_t+\varepsilon u$, $t\in [0,\infty)$, where $\varepsilon$ is a positive constant to be determined later. Then 8$_1$ becomes

Note that

Multiplying 33 by $U$ in $V_0$ and 8$_2$ by $v^t$ in $\mathcal{L}_{g,2}$, integrating over $\Omega$ and adding the two results, we obtain

where

and

Hence

where

Applying Cauchy's inequality with $\epsilon_1>0$, we get

It follows from 5 that, for any $\eta>0$, there exists a constant $C_\eta>0$ such that

Moreover,

and

Consequently, by taking sufficiently small $\epsilon_1$, $\eta$ and $\epsilon_2$ such that

we deduce that

Choosing

we obtain

and

Since

we conclude from 36 and the arguments similar to the proof of 16 that

It follows from 34 and 35 that

which yields

This, together with 37, gives

Hence $S(t)$ possesses an absorbing set with the radius $R>\sqrt{\frac{C_{14}+\varepsilon\varrho C_{17}}{\varepsilon\varrho C_{16}}(\|h\|^2+|\Omega|)}$.

Proof. We decompose $u = \hat{u}+\check{u}$ and $v^t = \hat{v}^t+\check{v}^t$ satisfying

where

Let $\psi = A^\delta \check{u}$, $\varphi^t = A^\delta \check{v}^t$, $0<\delta\leq \frac{1}{4}$. Then it follows from 39 that

Let $\Psi = \psi_t+\varepsilon \psi$, where $\varepsilon$ is a positive constant to be determined later. Then 40$_1$ can be written in the form

Multiplying 41 by $\Psi$ in $V_0$ and 40$_2$ by $\varphi^t$ in $\mathcal{L}_{g,2}$, integrating over $\Omega$ and adding the two results, we obtain

where

Hence

where

and

Note that

and

Consequently, taking

with some constant $0<\sigma_1<2$, we deduce from 42 that

We further choose

with some constant $0<\sigma_2<\sigma_1$. Thus

and

Applying Hölder's inequality and Cauchy's inequality to the right side of 43, we get

For any $w\in V_{1-4\delta}$, we have

Since

we deduce from 45 that

Combining this with 40$_3$ and 44, we obtain

Taking into account

we get

Let $\Upsilon = \bigcup\limits_{t\geq0}\check{v}^t$. Then $\Upsilon$ is bounded in $\mathcal{L}_{g,2+4\delta}\cap H_g^1(\mathbb{R}^+,V_{4\delta})$ due to 46, where $H_g^1(\mathbb{R}^+,V_{4\delta})$ is a Hilbert space of $V_{4\delta}$-valued functions $v$ on $\mathbb{R}^+$ such that both $v$ and $v_\tau$ belong to $\mathcal{L}_{g,4\delta}$. Note that $\sup_{v\in\Upsilon}\|v(\tau)\|_{V_2}^2\in L_g^1(\mathbb{R}^+)$. Hence, in view of [24,Lemma 5.5], we see that $\Upsilon$ is relatively compact in $\mathcal{L}_{g,2}$. Since $V_{2+4\delta}\times V_{4\delta}\hookrightarrow\hookrightarrow V_2\times V_0$, we conclude from 46 that there exists a $t_0 = t_0(B)$ such that $\bigcup\limits_{t\geq t_0}S_2(t)B$ is relatively compact. Thus the operators $S_2(t)$ are uniformly compact for $t$ large. Furthermore, as for problem 38, it is easy to check that

This means that $S_1(t)$ is a continuous mapping from $Z$ into itself such that

as $t\rightarrow\infty$. Therefore, by virtue of [26,Chapter I,Theorem 1.1] and Lemma 4.1, the proof of Theorem 2.3 is complete.

Acknowledgments

The author would like to thank the editors and the referees for the valuable comments and suggestions.