Attractors for a quasilinear viscoelastic equation with nonlinear damping and memory

1.   Introduction

In this paper we investigate the long-time dynamics of solutions for the quasilinear viscoelastic equation with nonlinear damping and memory

where $\Omega$ is a bounded domain of $\mathbb{R}^N$($N\geqslant 1$) with smooth boundary $\partial\Omega$, $u_0$ is the prescribed past history of $u$.

Problem (1.1) can be seen as an extension, accounting for memory effects in the material, of equations of the form

This equation is interesting not only from the point of view of PDE general theory, but also due to its applications in Mechanics. In the case $f(u_t)$ is a constant, Eq (1.2) has been used to model extensional vibrations of thin rods (see Love [1,Chapter 20]). In the case $f(u_t)$ is not a constant, Eq (1.2) can model materials whose density depends on the velocity $u_t$. We refer the reader to Fabrizio and Morro ^[2] for several other related models.

When $\rho = 0$ and $\Delta u_{tt}$ is dropped in (1.1), the related problem has been extensively studied and several results concerning the global existence, decay of global solution and finite time blow up have been established. In this direction, we refer the readers to see Ref. ^{[3,4,5,6,7,8,9,10,11,12]} and the references therein.

Now let us recall some results concerning quasilinear viscoelastic wave equations. In ^[13], Cavalcanti et al. studied the following equation with Dirichlet boundary conditions

A global existence result for $\gamma\geqslant 0$, as well as an exponential decay for $\gamma > 0$, has been established. This last result has been extended by Messaoudi and Tatar ^[14] to the case $\gamma = 0$.

In ^[15], Messaoudi and Tatar studied the following equation

By introducing a new functional and using a potential well method, they obtained the global existence of solutions and the uniform decay of the energy if the initial data are in some stable set. In the case $b = 0$ in (1.4), Messaoudi and Tatar ^[16] proved the exponential decay of global solutions to (1.4), without smallness of initial data, considering only the dissipation effect given by the memory. Liu ^[17] proved that for certain initial data in the stable set, the solution decays exponentially, and for certain initial data in the unstable set, the solution blows up in finite time.

Replacing strong damping by weak damping in Eq (1.3), several authors have studied the energy decay rates of the related problems like

When $h(u_t) = u_t$, Han and Wang ^[18] investigated the global existence and exponential stability of the energy for solutions for Eq (1.5). When $h(u_t) = |u_t|^mu_t(m > 0)$, the same authors ^[19] proved the general decay of energy for Eq (1.5). Later, Park and Park ^[20] established the general decay for Eq (1.5) with general nonlinear weak damping.

Now, we list some important literature on the nonlinear evolution equation with hereditary memory and variable density. Araújo et al. ^[21] studied the following equation

and proved the global existence, uniqueness and exponential stability of solutions and existence of the global attractor. Subsequently, Qin et al. ^[22,23] proved the upper semicontinuity of pullback attractors and the existence of uniform attractors by assuming $f(u) = 0$ and taking a frictional damping $u_t$ instead of strong damping $-\Delta u_t$. However, their argument for uniqueness rely on the differentiability of the map $\sigma(x) = |x|^{\rho}$ at zero, which introduces the further restriction $\rho > 1$.

Lately, the authors ^[24] established an existence, uniqueness and continuous dependence result for the weak solutions to the semigroup generated for the system (1.6) in a three-dimensional space when $\rho\in [0, 4]$ and $f$ has polynomial growth of (at most) critical order $5$. Then, based on the ^[24], the same authors ^[25] established the existence of the global attractor of optimal regularity for Eq (1.6) when $\rho\in[0, 4)$. Recently, the authors ^[26] proved that the sole weak dissipation $(\gamma = 0)$ given by the memory term is enough to ensure existence and optimal regularity of the global attractor. Leuyacc and Parejas ^[27] proved the upper semicontinuity of global attractors when $\rho\rightarrow0^+$ in (1.6). Li and Jia ^[28] proved the existence of a global solution by means of the Galerkin method, establish the exponential stability result and the polynomial stability result when the kernel $\mu(s)$ satisfies $\mu'(s)\leqslant-k_1\mu^q(s), \ 1\leqslant q < 3/2$.

Motivated by the works above mentioned, our aim is to present the existence of global attractors for the problem (1.1).

As in^[29,30,31], we shall introduce a new variable $\eta^t$ to the system which corresponds to the relative displacement history.

Let us define

Note that

Thus, the original memory term can be rewritten as

and Eq (1.1) becomes

Assuming for simplicity that $\alpha-\int_0^{\infty}\mu(s){\rm d}s = 1$, we have the new system

where

2.   Assumptions and the main result

We begin with precise hypotheses on problem (1.7). Assume that

Concerning the source term $f:\mathbb{R}\rightarrow\mathbb{R}$, we assume that

where $c_0 > 0$ and

In addition, we assume that

where $F(u) = \int_0^uf(s)ds$.

The damping function $g\in \mathcal{C}^1(\mathbb{R})$ is a non-decreasing function with $g(0) = 0$ and satisfies the polynomial condition

where $c_1 > 0$ and

With respect to the memory component, we assume that

and there exist $k_0, k_1 > 0$ such that

and

As usual, $\|\cdot\|_p$ denotes the $L^p$-norms as well as $(\cdot, \cdot)$ denotes either the $L^2$-inner product. Let $\lambda_1 > 0$ be the first eigenvalue of $-\Delta$ in $H_0^1(\Omega)$.

In order to consider the relative displacement $\eta^t$ as a new variable, one introduces the weighted $L^2-$space

which is a Hilbert space endowed with inner product and norm

respectively.

Next let us introduce the phase space

endowed with the norm

Then, the energy of problem (1.7) is given by

According to the arguments ^[24] with slightly modified, we can obtain the following well-posedness result.

Theorem 2.1. Assume the assumptions (2.1)-(2.7) hold. If initial data $z_0 = (u_0, u_1, \eta_0)\in\mathcal{H}$ and $h\in L^2(\Omega)$, then the problem (1.7) admits a unique global solution

satisfying

Remark 1. The well-posedness of problem (1.7) given by Theorem 2.1 implies that the one-parameter family of operators $S(t):\mathcal{H}\rightarrow\mathcal{H}$ defined by

where $z = (u(t), u_t(t), \eta^t(t))$ is the unique weak solution of the system (1.7), satisfies the semigroup properties

and defines a nonlinear $C_0$-semigroup. Then problem (1.7) can be viewed as a nonlinear infinite dynamical system $(\mathcal{H}, S(t))$.

Now we give the following result concerning the global attractors.

Theorem 2.2. Assume the assumptions (2.1)-(2.7) hold and $h\in L^2(\Omega)$. Then the dynamical system $(\mathcal{H}, S(t))$ generated by (1.7) has a compact global attractor $\mathcal{A}\subset \mathcal{H}$.

3.   Global attractors

Before presenting our results we recall some fundamentals of the theory of infinite-dimensional dynamical systems which can be founded in the book by Chueshov and Lasiecka ^[32,33].

Theorem 3.1. A dissipative dynamical system $(\mathcal{X}, S(t))$ has a compact global attractor if and only if it is asymptotically smooth.

The proof of asymptotic smoothness property can be very delicate. Here we use the following “compensated compactness” result ^[32,34].

Theorem 3.2. Let $(\mathcal{X}, S(t))$ be a dynamical system on a complete metric space $\mathcal{X}$ endowed with a metric $d$. Suppose that for any bounded positively invariant set $B\subset \mathcal{X}$ and for any $\varepsilon > 0$, there exists $T = T(\varepsilon, B)$ such that

where $\Phi_T:B\times B\rightarrow\mathbb{R}$ satisfies

for any sequence $\{z_n\}_{n\in\mathbb{N}}$ in $B$. Then $S(t)$ is asymptotic smooth in $\mathcal{X}$.

In the sequel we will apply the abstract results presented above to prove Theorem 2.2. Firstly, we show that the dynamical system $(\mathcal{H}, S(t))$ is dissipative. The next step is to verify the asymptotic smoothness. Then the existence of a compact global attractor is guaranteed by Theorem 3.1. In what follows, the generic positive constants will be denoted as $C$, while $\mathcal{Q}(\cdot)$ will stand for a generic increasing positive function.

3.1.   Existence of an absorbing set

In this section, our aim is to show that the dynamical system $(\mathcal{H}, S(t))$ is dissipative. To this aim, we first give some priori estimates used later.

Proposition 3.1. For any initial data $z_0$ with $\|z_0\|_{\mathcal{H}}\leqslant R$, we have the uniform estimate

Proof. Multiplying (1.7) by $(u_t, \eta^t)$, we obtain

Owing to (2.7), one can easily see that

Combining (3.3) and (3.2), we have

Since $\mu(s)$ is decreasing, $g'\geqslant 0$ and $g(0) = 0$, we get

and consequently

Applying Young inequality yields

It follows promptly from (2.10) that

Then making use of (2.4) we obtain

This means that

A multiplication of (1.7) by $u_{tt}$ gives

Next, we estimate each term individually. By Hölder inequality, Poincaré inequality and Young inequality, we have

and

Using Hölder inequality, Poincaré inequality, Young inequality, and (3.5), we have

and

Substituting all the above inequalities into (3.6) and taking (3.5) into account, we derive that

This means that

In light of (3.5) and (3.7), we obtain the Proposition 3.1.

Now, we introduce the following two functionals

and

Lemma 3.3. There exists a positive constant $C_1$, such that

Proof. A multiplication of the first equation of (1.7) by $u$ gives

By Hölder inequality and Cauchy inequality, we obtain

Using Hölder inequality, Young inequality and Sobolev inequality, taking into account (2.5) and (2.6), we arrive at

Combining the last two estimates, we end up with

Besides, in light of (2.9), we get

Using (2.10), (2.4) and (3.11) yields

This lemma is complete.

Lemma 3.4. For any $\delta_1 > 0$, there exists $C_2 > 0$ such that

Proof. Taking the time derivative of $\Psi(t)$, in light of the first equation of (1.7), we get

Next, we will estimate the right side of the above identity. Integrating by parts with respect to $x$ and using Young inequality, we obtain

and

Applying (2.5), Hölder inequality, Sobolev embedding inequality, Young inequality and Proposition 3.1, we obtain

Analogously, but using (2.2) instead of (2.5), we have

Using Hölder inequality, Young's inequality and Sobolev embedding inequality, we get

On the other hand, since

we find

Applying Hölder inequality, Young's inequality and Sobolev embedding inequality, we obtain

and

Collecting all the above inequalities and (3.11), we end up with

This lemma is complete.

Lemma 3.5 (Absorbing set). Under the hypotheses of Theorem 2.2, the dynamical system $(\mathcal{H}, S(t))$ corresponding to problem (1.7) has a bounded absorbing set.

Proof. Let us consider a perturbed functional

where $\Phi(t)$ and $\Psi(t)$ are the same functional defined in (3.8) and (3.9).

Since

we get

Then making use of (2.4) we obtain

Now we claim that there exist three constants $\alpha_1, \alpha_2, b > 0$ such that

To prove this, we first note that

Hence there exists a constant $b > 0$ such that

Then taking $M > b$, we get (3.15) with $\alpha_1 = M-b$ and $\alpha_2 = M-b$.

Combining (3.4), (3.13), Lemma 3.3 and 3.4, we arrive at

Choosing $\varepsilon, \delta_1 > 0$ small enough and $M > 0$ sufficiently large such that

then we have

which together with (3.15) implies that

Integrating the above inequality over $[0, t]$, we can derive

Using again (3.15) yields

Recalling (3.14), we obtain

where

This shows that any closed ball $\bar{B}(0, R)$ with $R > R_0$ is a bounded absorbing set of $(\mathcal{H}, S(t))$. The proof of Lemma 3.5 is now complete.

As a straight consequence of Lemma 3.5, we have that the solutions of problem (1.7) are globally bounded provided initial data lying in bounded sets $B\subset\mathcal{H}$. Namely, let $z = (u, u_t, \eta^t)$ be a solution of (1.7) with initial data $z_0 = (u_0, u_1, \eta_0)$ in a bounded set $B$. Then one has

where $C_B$ is a constant depending on $B$. Lemma 3.5 also ensures the existence of bounded positively invariant sets.

3.2.   A stability inequality

Lemma 3.6 (Stabilizability). Under the hypotheses of Theorem (2.2), given a bounded set $B\subset\mathcal{H}$, let $z_1 = (u, u_t, \eta^t)$ and $z_2 = (v, v_t, \xi^t)$ be two weak solutions of problem (1.7) such that $z_1(0) = (u_0, u_1, \eta_0)$ and $z_2(0) = (v_0, v_1, \xi_0)$ are in $B$. Then

where $w = u-v$ and $\gamma, C_B$ are positive constants depending on $B$.

Proof. Let us write $w = u-v$ and $\zeta^t = \eta^t-\xi^t$. Then $(w, \zeta^t)$ satisfies

Now we consider the functional

and its perturbation

where

We divide the remaining of the proof into five steps. Hereafter, we use $C_B$ to denote several positive constants.

Step 1. For $M > 0$ sufficiently large, there exists $\varepsilon_0$ such that

To prove this, we first observe that

Besides, using Hölder inequality, Sobolev inequality, Young inequality and (3.16), we can derive that

Collecting the above two estimates and (3.19), we obtain

Now let us put $\varepsilon_0 = M/2-C_B$. Then for all $\varepsilon\leqslant\epsilon_0$, the inequality (3.20) holds.

Step 2. There exists a constant $C_3 > 0$ such that

In fact, differentiating (3.18) with respect to $t$ and using (3.17), we have

Using Hölder inequality, Sobolev inequality, estimate (3.7) and (3.16), we have

Combining (2.2), Hölder inequality, Sobolev inequality, and estimate (3.16) yields

Since $g$ is a non-decreasing function, this yields

Inserting last three estimates into (3.22), we arrive at

Step 3. There exists $C_4 > 0$ such that

Taking the derivative of $\phi(t)$, it follows from (3.17) that

From Hölder inequality, Sobolev embedding, estimate (3.7) and (3.16), we have

Applying (2.2), Hölder inequality, Sobolev embedding, and estimate (3.16), we get

Similarly, in light of (2.5), Hölder inequality, Young inequality, Sobolev embedding, and estimate (3.16), we obtain

Using Young inequality gives

Combining these six last estimates with (3.18) we end up with

which together with (3.11) implies that inequality (3.23) holds for some $C_4 > 0$.

Step 4. For any $\delta_2 > 0$, there exists $C_5 > 0$ such that

Taking derivative of $\psi(t)$ and using (3.17), we derive that

Applying Hölder inequality, Young inequality, Sobolev embedding, estimate (3.7) and (3.16), we get

From (3.17), one can easily see that

Hence in light of Young inequality we obtain

By the monotonicity of function $x\mapsto|x|^{\rho}x\ (\rho > 0)$, we get

Using Hölder inequality, Young inequality and estimate (3.16), we obtain

Inserting (3.27)-(3.32) into (3.26), (3.11) we end up with

Step 5. Combining (3.21), (3.23), (3.25) with (3.19), we have

Firstly we fix $\varepsilon > 0$ such that $\varepsilon C_4 < k_0/2$. Then taking $\delta_2 > 0$ such that $\delta_2 < \varepsilon/8$. For fixed $\varepsilon$ and $\delta_2$, we choose $M > 0$ so large that $M > 2(\varepsilon C_4+C_5)$. Then (3.33) along with (3.20) give

Integrating (3.34) over $(0, t)$ with respect to $t$, we get

which together with (3.20) implies that

where $\gamma = 2\varepsilon/3M$ is a positive constant. Notice that the functional $E_w(t)$ is equivalent to the norm of $\mathcal{H}$, the proof is complete.

3.3.   Existence of global attractor

Lemma 3.7 (Asymptotic smoothness). Under the hypotheses of Theorem 2.2, the dynamical system corresponding to problem (1.7) is asymptotic smooth.

Proof. Let $B$ be a bounded subset of $\mathcal{H}$ positively invariant with respect to $S(t)$. Let $S(t)z_1(0) = (u, u_t, \eta^t)$ and $S(t)z_2(0) = (v, v_t, \xi^t)$ be two solutions for problem (1.7) corresponding to initial data $z_1(0), z_2(0)\in B$. Given $\varepsilon > 0$, we can choose $T > 0$ so large that $C_Be^{-\gamma t} < \varepsilon$. We claim that there exists constant $C_{BT} > 0$ such that

with

Indeed, from Lemma 3.6, we have

and consequently (3.35) and (3.36) hold.

We are left to prove that $\Phi_T$ satisfies (3.1). Indeed, given a sequence of initial data $z_n = (u_0^n, u_1^n, \eta_0^n)\in B$, we write $S(t)z_n = (u^n(t), u_t^n(t), \eta^{n, t})$. Since $B$ is invariant by $S(t), \ t\geqslant 0$, it follows that $(u^n(t), u_t^n(t), \eta^{n, t})$ uniformly bounded in $\mathcal{H}$. Namely,

Then by compact embedding $H_0^1(\Omega)\hookrightarrow L^{\rho+2}(\Omega)$ and $H_0^1(\Omega)\hookrightarrow L^{p+2}(\Omega)$, there exists a subsequence $(u^{n}, u_t^{n}, \eta^{n, t})$ such that

Therefore,

which implies (3.1) holds. Then asymptotic smoothness follows from Theorem 3.2.

Proof of Theorem 2.2. We first note that Lemmas 3.5 and 3.7 imply that $(\mathcal{H}, S(t))$ is a dissipative dynamical system which is asymptotically smooth. Then the existence of a compact global attractor $\mathcal{A}$ to problem (1.7) in the phase space $\mathcal{H}$ follows from Theorem 3.1.

Acknowledgments

The authors are grateful to the referees for the constructive comments and kind suggestions.

Conflict of interest

All authors declare no conflicts of interest in this paper.