Skip to content

Advertisement

  • Research
  • Open Access

Long-time behavior of a semilinear wave equation with memory

Boundary Value Problems20162016:37

https://doi.org/10.1186/s13661-016-0551-5

  • Received: 8 January 2016
  • Accepted: 31 January 2016
  • Published:

Abstract

In this paper we study the long-time dynamics of the semilinear viscoelastic equation
$$u_{tt} - \Delta u_{tt} - \Delta u + \int_{0}^{\infty} \mu(s) \Delta u(t-s) \,ds + f(u) = h , $$
defined in a bounded domain of \(\mathbb{R}^{3}\) with Dirichlet boundary condition. The functions \(f=f(u)\) and \(h=h(x)\) represent forcing terms and the kernel function \(\mu\ge0\) is assumed to decay exponentially. Then, by exploring only the dissipation given by the memory term, we establish the existence of a global attractor to the corresponding dynamical system.

Keywords

  • wave equation
  • global attractor
  • memory
  • viscoelasticity

MSC

  • 35L71
  • 35B41
  • 74D99

1 Introduction

This is paper is concerned with the long-time behavior of a class of wave equations with memory of the form
$$\begin{aligned}& u_{tt} - \Delta u_{tt} - \Delta u + \int_{0}^{\infty} \mu(s) \Delta u(t-s) \,ds + f(u) = h \quad \mbox{in } \Omega\times\mathbb{R}^{+}, \end{aligned}$$
(1.1)
$$\begin{aligned}& u =0 \quad \mbox{on } \Gamma\times\mathbb{R}^{+}, \end{aligned}$$
(1.2)
with initial conditions
$$ \begin{aligned} &u(x,0)=u_{0}(x), \qquad u_{t}(x,0)=v_{0}(x),\quad \forall x \in\Omega, \\ &u(x,-s)= \varphi(x,s), \quad \forall (x,s) \in\Omega\times\mathbb {R}^{+} , \end{aligned} $$
(1.3)
where Ω is a bounded domain in \(\mathbb{R}^{3}\) with smooth boundary Γ, and φ is a prescribed past history.
This problem is related to a model of extensional vibrations of thin rods
$$u_{tt} -\Delta u_{tt} - \Delta u =0, $$
described in Love [1], Chapter 20, which is a conservative system. Here, we have added a nonlinear forcing \(f(u)\) and a dissipation of memory type.
We observe that such a system was extensively studied in the more general form
$$ |u_{t}|^{\rho} u_{tt} - \Delta u_{tt} - \Delta u + \int_{0}^{\infty} \mu (s) \Delta u(t-s) \,ds + f(u) = 0, $$
(1.4)
with \(0 \le\rho< 2\). Most of results are concerned with the exponential stability of the system under additional damping \(-\Delta u_{t}\) or \(u_{t}\). We refer the reader to, e.g., [25]. The existence of global attractors to (1.4) was first proved in Araújo et al. [6], with the assumption \(\rho>1\) and with the additional damping \(-\Delta u_{t}\). The assumption \(\rho>1\) was technical and related to the uniqueness of the problem. Later, it was shown in [7] that the strong damping \(-\Delta u_{t}\) could be replaced by the weak damping \(u_{t}\), but yet with \(\rho> 1\). On the other hand, in [8], the existence of a global attractor for the problem with \(\rho=\mu=0\) was studied with a strong damping.

More recently, it was proved by Conti et al. [9] that existence and uniqueness for the mixed problem (1.4) holds for \(\rho\ge0\) and without additional damping terms, that is, keeping only the dissipation given by the memory. This means that the restriction \(\rho>1\) can be dropped.

Motivated by results in [6] and [9], we propose to study the existence of global attractors of (1.4) with \(\rho=0\) and exploring only the dissipation given by the memory term. That is, we consider the problem (1.1)-(1.3). Then our result extends or complements the ones in [69]. See Theorem 3.1.

Of course, if the rotational inertia \(\Delta u_{tt}\) is dropped, then equation (1.1) becomes the well-known viscoelastic wave equation of memory type. On this matter, we refer the reader to some relevant results in [1014], among others.

2 History setting

We denote by \((\cdot, \cdot)\) and \(\|\cdot\|\) is the standard inner product and norm on \(L^{2}(\Omega)\). It is well known that the operator A with domain \(D(A)\) defined by
$$A=-\Delta, \qquad D(A)=H^{2}(\Omega) \cap H_{0}^{1}( \Omega), $$
is self-adjoint and strictly positive. See, e.g., [15]. We adopt the notation
$$H_{0}= L^{2}(\Omega), \qquad H_{1}=H_{0}^{1}( \Omega) \quad \mbox{and} \quad H_{2}= H^{2}(\Omega)\cap H_{0}^{1}(\Omega). $$
Next we establish the history setting of the problem (1.1)-(1.3) in order to deal with the non-autonomous character of the memory term in (1.1). We follow the arguments of [9, 12, 14], based on [16]. Let \(\mu:\mathbb{R}^{+} \to[0, \infty)\) be a summable function. We denote by \(\mathcal{M}\) the \(L^{2}\)-weighted space defined by
$$ L^{2}_{\mu}\bigl(\mathbb{R}^{+};H_{1}\bigr)= \bigl\{ \eta:\mathbb{R}^{+} \to H_{1}: \|\eta \|_{\mathcal{M}} < \infty\bigr\} , $$
where \(\| \cdot\|_{\mathcal{M}} = (\cdot,\cdot)_{\mathcal{M}}^{\frac {1}{2}}\) and \((\eta,\xi)_{\mathcal{M}} = \int_{0}^{\infty}\mu(\tau)(\eta (\tau),\xi(\tau))_{1}\, d\tau\). Similarly we define the space \(\mathcal {M}_{0}\) as
$$ L^{2}_{\mu}\bigl(\mathbb{R}^{+};H_{0}\bigr)= \bigl\{ \eta:\mathbb{R}^{+} \to H_{0}: \|\eta \|_{\mathcal{M}_{0}} < \infty\bigr\} , $$
where \(\| \cdot\|_{\mathcal{M}_{0}} = (\cdot,\cdot)_{\mathcal {M}_{0}}^{\frac{1}{2}}\) and \((\eta,\xi)_{\mathcal{M}_{0}} = \int_{0}^{\infty}\mu(\tau)(\eta(\tau),\xi (\tau))_{0} \, d\tau\). From classical theory, the spaces \(\mathcal{M}\) and \(\mathcal{M}_{0}\) are separable Hilbert spaces.
Let T be the infinitesimal generator of the right-translation semigroup on \(\mathcal{M}\), that is,
$$T \eta=-\eta', $$
for all \(\eta\in D(T)=\{ \eta\in\mathcal{M}: \eta' \in\mathcal {M}, \eta(0)=0 \}\), where \(\eta'(t) = \frac{\partial \eta}{\partial t}\) in the sense of distributions and \(\eta (0)= \lim_{s \to0}\eta(s)\). It is well known that
$$( T \eta, \eta)_{\mathcal{M}} \leq0, \quad \forall \eta\in D(T). $$
We also introduce the Hilbert space
$$\mathcal{H}= H_{1} \times H_{1} \times\mathcal{M}, $$
endowed with the norms \(\|\cdot\|_{\mathcal{H}}=(\cdot,\cdot)_{\mathcal{H}}^{\frac{1}{2}}\) where
$$\bigl((u_{1},v_{1},\eta_{1}), (u_{2},v_{2}, \eta_{2})\bigr)_{\mathcal{H}}= (u_{1}, u_{2})_{1} + (v_{1}, v_{2})_{1} + (\eta_{1}, \eta_{2})_{\mathcal{M}}. $$
Then, as in [12, 14], we define
$$\eta= \eta^{t}(x,s)= u(x,t) - u(x,t-s), \quad s \in\mathbb{R}^{+}. $$
Using this new variable η we can reformulate the system (1.1)-(1.3) to become
$$ \left \{ \textstyle\begin{array}{l} u_{tt} + Au_{tt} + Au - \int_{0}^{\infty} \mu(s) A\eta(s) \,ds + f(u) = h, \\ \eta_{t}= T\eta+ u_{t}, \end{array}\displaystyle \right . $$
(2.1)
with initial conditions
$$ u(0)=u_{0},\qquad u_{t}(0)=v_{0} ,\qquad \eta^{0}(s)= \eta_{0}, $$
(2.2)
where \(\eta_{0}(s) = u_{0} - \varphi(s)\) for all \(s \in\mathbb{R}^{+}\).
The system (2.1)-(2.2) is a particular case of the system considered in [9]. There, the authors established the well-posedness for a class of problems with \(|u_{t}|^{\rho}u_{tt}\) instead of \(u_{tt}\) as in (2.1). They proved, among other results, that the system (2.1)-(2.2) with initial data \(z=(u_{0},v_{0},\eta_{0}) \in\mathcal{H}\) admits a unique weak solution
$$(u,\eta) \in W^{2,\infty}(0,\tau;H_{1}) \times C\bigl([0,\tau]; \mathcal{M}\bigr), $$
satisfying the identity
$$ ( u_{tt}, \phi) + (u_{tt}, \phi)_{1} + (u, \phi)_{1} + \int_{0}^{\infty} \mu(s) \bigl( \eta(s), \phi \bigr)_{1} \, ds + \bigl(f(u), \phi\bigr) = (h, \phi), $$
(2.3)
for every \(\phi\in H_{1}\) and for a.e. \(t>0\). Here, η is a mild solution to the non-homogeneous linear equation in the Hilbert space \(\mathcal{M}\),
$$\frac{d}{dt}\eta= T \eta+ u_{t}, $$
where τ is a positive real number arbitrarily fixed. In addition, it was shown in Estimate (4.4) of [9],
$$ \| u_{tt} \|_{1} \le C , \quad \mbox{a.e. } t \in[0, \tau], $$
(2.4)
where \(C>0\) depends only on the initial data.
Now, due to the continuous dependence on initial data, the weak solution \((u,\eta)\) of the system (2.1)-(2.2), with initial data \((u(0),u_{t}(0),\eta^{0})=z\), can be rewritten in the form
$$ S(t)z=\bigl(u(t),u_{t}(t),\eta^{t}\bigr), $$
(2.5)
generating a \(C_{0}\)-semigroup \(S(t)\) on \(\mathcal{H}\).
We end this section by recalling that a global attractor for a \(C_{0}\)-semigroup \(S(t)\) on \(\mathcal{H}\) is a compact subset \(\mathbf {A}\subset\mathcal{H}\) which is strictly invariant, that is, \(S(t)\mathbf{A} = \mathbf{A}\), \(\forall t \ge0\), and uniformly attracting, that is,
$$\operatorname{dist}_{\mathcal{H}}\bigl(S(t){B},\mathbf{A}\bigr) = \sup _{x\in S(t){B}} \inf_{y\in \mathbf{A}}\|x-y\|_{\mathcal{H}} \to0 \quad \mbox{as } t \to \infty, $$
for any bounded set \({B}\subset\mathcal{H}\).

3 Global attractors

In this section we establish our main result. The assumptions we make in this paper are as follows.
(H1): 
Assume \(\mu\in C^{1}(\mathbb{R}^{+}) \cap L^{1}(\mathbb {R}^{+})\) and satisfying the following conditions:
  1. (i)

    \(\mu(s) \geq0\) for all \(s \in\mathbb{R}^{+}\);

     
  2. (ii)

    there exists a positive constant \(k_{1}\) such that \(\mu'(s) \leq- k_{1}\mu(s)\) for all \(s \in\mathbb{R}^{+}\).

     
(H2): 
The nonlinearity \(f \in C^{1}(\mathbb{R})\) and verifies the following conditions:
  1. (i)

    \(|f(r) - f(s)| \leq C (1 + |r|^{p} + |s|^{p})|r -s|\) for all \(r,s \in\mathbb{R}\), where \(0 \leq p <4\);

     
  2. (ii)

    there exists a positive constant ρ such that \(f(s) s - \hat{f}(s) \geq- \rho\), where \(\hat{f}(s) = \int_{0}^{s} f(\tau)\, d\tau\).

     
(H3): 

The forcing h belongs to the dual space of \(H_{1}\).

To simplify the notation we write \(\|\mu\|_{L^{1}(\mathbb{R}^{+})} = k_{0}\) in our estimates. We also observe that the energy associated with the problem (2.1)-(2.2) is given by
$$\begin{aligned} E(t) =& \frac{1}{2}\|u_{t}\|^{2} + \frac{1}{2} \|u_{t}\|_{1}^{2} + \frac{1}{2}\|u \|_{1}^{2} + \frac{1}{2}\|\eta\|_{\mathcal{M}}^{2} \\ &{}+ \int _{\Omega}\hat{f}(u) \, dx - \int_{\Omega} h u \, dx,\quad t \geq0. \end{aligned}$$
Our main result is the following.

Theorem 3.1

Suppose that the conditions (H1)-(H3) are verified. Then the dynamical system \((S(t),\mathcal{H})\) generated by the problem (2.1)-(2.2) has a global attractor.

The proof of this theorem will be completed at the end of this section.

3.1 Abstract theory

Let us present a small collection of well-known results of from the theory of attractors. This can be found in, e.g., [1721]. A dynamical system \((\mathcal{H},S(t))\) is called dissipative if the semigroup \(S(t)\) has an absorbing set, that is, a bounded set \({B}_{0} \subset\mathcal{H}\) such that
$$S(t){B} \subset{B}_{0} , \quad \forall t \geq t_{{B}}, $$
for all bounded set \({B} \subset\mathcal{H}\). A semigroup \(S(t)\) is asymptotically smooth in \(\mathcal{H}\) if for any bounded positive invariant set \({B} \subset\mathcal{H}\), that is, \(S(t){B} \subseteq {B}\) for all \(t\geq0\), there exists a compact set \(K \subset\overline {B}\), such that
$$\operatorname{dist}_{\mathcal{H}}\bigl(S(t){B},K\bigr) \to0 \quad \mbox{as } t \to\infty. $$
Then a classical result asserts that a dissipative \(C_{0}\)-semigroup \(S(t)\) defined on \(\mathcal{H}\) has a compact global attractor in \(\mathcal{H}\) if and only if it is asymptotically smooth in \(\mathcal{H}\).
Now, it is well known from Proposition 2.10 in [18] that a \(S(t)\) is asymptotically smooth in \(\mathcal{H}\) if for any positively invariant set \({B} \subset\mathcal{H}\), and for any \(\varepsilon>0\), there exists \(T=T({\varepsilon,B})\) such that
$$ \bigl\Vert S(T)x -S(T)y\bigr\Vert _{\mathcal{H}} \leq \varepsilon+ \phi_{T}(x,y),\quad \forall x,y \in{B}, $$
(3.1)
where \(\phi_{T}: {B} \times{B} \to\mathbb{R}\) satisfies
$$ \liminf_{n \to\infty} \liminf_{m \to\infty} \phi_{T}(z_{n},z_{m})=0, $$
(3.2)
for any sequence \((z_{n})\) in B.

3.2 Dissipativeness

In this section we shall construct a bounded absorbing set to our system \((\mathcal{H},S(t))\) where \(S(t)\) is the solution operator defined in (2.5). Let \((u,u_{t},\eta)\) be a weak solution of the system (2.1)-(2.2). Since \(H_{1}\) is dense in \(H_{2}\) and \(u \in L^{\infty}(0,\tau;H_{1})\) we can assume that u is more regular and obtain
$$\|u\|_{L^{p+2}}^{p+2}\leq K_{\Omega}\|u\|_{1}^{p+2} \|u\|, $$
where \(K_{\Omega}\) is an embedding constant. To simplify the notation denote all the embedding constants by \(C_{\Omega}\). Replacing ϕ by \(u_{t}\) in (2.3) and adding the identity \((\eta_{t}, \eta ) = (T \eta, \eta)_{\mathcal{M}} + (u_{t}, \eta)_{\mathcal{M}}\) we get
$$\frac{d}{dt} E(t) = (T\eta,\eta)_{\mathcal{M}} = \frac{1}{2} \int _{0}^{\infty} \mu'(s)\bigl\Vert \eta(s)\bigr\Vert _{1}^{2} \, ds \leq0. $$
We are going to apply the perturbed energy method. We consider the maps
$$ \Psi(t)=\bigl(u_{t}(t),u(t)\bigr) + \bigl(u_{t}(t),u(t)\bigr)_{1}, \qquad \Phi(t) = \bigl(u_{t}(t),\eta ^{t}\bigr)_{\mathcal{M}} + \bigl(u_{t}(t),\eta^{t}\bigr)_{\mathcal{M}_{0}} $$
(3.3)
and
$$ F_{\varepsilon}(t) = \varepsilon^{-1}E(t) + \varepsilon\Psi(t) - \Phi(t), $$
(3.4)
where \(0 < \varepsilon\leq1\). Then we have the following result.

Lemma 3.2

Let Ψ, Φ and \(F_{\varepsilon}\) be the maps defined in (3.3) and (3.4). Then
  1. (a)
    there exist constants \(\beta=\beta(k_{0},\lambda_{1}) \geq1\) and \(C_{\rho,\lambda_{1},h,\Omega} \geq0\) such that
    $$\bigl\vert \varepsilon\Psi(t) - \Phi(t)\bigr\vert \leq\beta\bigl(E(t) + C_{\rho,\lambda _{1},h,\Omega}\bigr); $$
     
  2. (b)
    for every \(0< \varepsilon< \beta^{-1}\), the positive constants \(\beta_{1}= \varepsilon^{-1}-\beta\) and \(\beta _{2}=\varepsilon^{-1}+\beta\) satisfy the inequality
    $$\beta_{1} E(t) - \frac{1}{2} C_{\rho,\lambda_{1},h,\Omega} \leq F_{\varepsilon}(t) \leq\beta_{2} E(t) + \frac{1}{2} C_{\rho,\lambda _{1},h,\Omega}. $$
     

Proof

By using Sobolev embeddings and condition (H2) we have
$$\begin{aligned} \biggl\vert - \int_{\Omega} \bigl( \hat{f}(u) - hu\bigr) \,dx \biggr\vert = & \biggl\vert - \int_{\Omega} \bigl( \hat{f}(u) - f(u)u + f(u)u - hu\bigr) \,dx \biggr\vert \\ \leq& \biggl\vert \int_{\Omega} \bigl( \hat{f}(u) - f(u)u\bigr) \,dx \biggr\vert + \biggl\vert \int_{\Omega} \bigl(f(u)u - hu\bigr) \,dx \biggr\vert \\ \leq& \rho \vert \Omega \vert + \int_{\Omega} \bigl\vert \bigl(f(u)u - hu\bigr) \bigr\vert \,dx \\ \leq& \rho \vert \Omega \vert + \int_{\Omega} \bigl( 1 + \vert u\vert ^{q}\bigr) \vert u\vert ^{2}\, dx + 6 \lambda_{1}^{-1} \|h\|^{2} + \frac{1}{12}\|u\|_{1}^{2} \\ \leq& \rho \vert \Omega \vert + \|u\|^{2} + \|u \|_{p+ 2}^{p + 2} + 6\lambda _{1}^{-1}\|h \|^{2} + \frac{1}{12}\|u\|_{1}^{2} \\ \leq& \rho \vert \Omega \vert + K_{\Omega}\|u\|_{1} + K_{\Omega}\|u\|_{1} + 6\lambda_{1}^{-1}\|h \|^{2} + \frac{1}{12}\|u\|_{1}^{2} \\ \leq& \rho \vert \Omega \vert + 12(K_{\Omega})^{2} + \frac{1}{12}\|u\|_{1}^{2} + \frac{1}{12}\|u \|_{1}^{2} \\ &{} + 6\lambda_{1}^{-1}\|h\|^{2} + \frac{1}{12}\|u\|_{1}^{2}, \end{aligned}$$
(3.5)
where \(|\Omega|\) is the measure of Ω. Therefore, there is a positive constant \(C_{\rho,\lambda_{1},h,\Omega}\) such that
$$ \frac{1}{4}\|u\|_{1}^{2} + \int_{\Omega} \bigl( \hat{f}(u) - hu\bigr) \,dx + C_{\rho,\lambda_{1},h,\Omega} \geq0. $$
(3.6)
Applying the inequality (3.6) we obtain
$$\begin{aligned} \bigl\vert \varepsilon\Psi(t) - \Phi(t)\bigr\vert \leq& \varepsilon\bigl\vert (u_{t},u) + (u_{t},u)_{1}\bigr\vert + \bigl\vert (u_{t},\eta)_{\mathcal{M}} + (u_{t}, \eta)_{\mathcal{M}_{0}}\bigr\vert \\ \leq& \varepsilon \biggl(2\lambda_{1}^{-1} \|u_{t}\|^{2} + \frac{1}{8}\|u\| _{1}^{2} \biggr) + \varepsilon \biggl(2\|u_{t}\|_{1}^{2} + \frac{1}{8}\|u \| _{1}^{2} \biggr) \\ &{} + \biggl( k_{0}\|u_{t}\|_{1}^{2} + \frac{1}{4}\|\eta\|_{\mathcal {M}}^{2} \biggr) + \biggl(k_{0}\|u_{t}\|^{2} + \frac{1}{4}\|\eta \|_{\mathcal {M}}^{2} \biggr) \\ = & \bigl(k_{0} + 2\lambda_{1}^{-1}\varepsilon \bigr) \|u_{t}\|^{2} + (k_{0} + 2\varepsilon) \|u_{t}\|_{1}^{2} + \frac{\varepsilon}{4} \|u \|_{1}^{2} + \frac {1}{2} \|\eta\|_{\mathcal{M}}^{2} \\ \leq& \bigl(k_{0} + 2\lambda_{1}^{-1}\varepsilon \bigr) \|u_{t}\|^{2} + (k_{0} + 2\varepsilon) \|u_{t}\|_{1}^{2} + \frac{\varepsilon}{4} \|u \|_{1}^{2} + \frac {1}{2} \|\eta\|_{\mathcal{M}}^{2} \\ &{} + \biggl( \frac{1}{4}\|u\|_{1}^{2} + \int_{\Omega} \bigl( \hat{f}(u) - hu\bigr) \,dx + C_{\rho,\lambda_{1},h,\Omega} \biggr) \\ \leq& \beta\bigl(E(t) + C_{\rho,\lambda_{1},h,\Omega}\bigr), \end{aligned}$$
where \(\beta= \sup_{0 <\varepsilon\leq1} \{1,\varepsilon ,2(k_{0} + 2\varepsilon), 2(k_{0} + 2\lambda_{1}^{-1} \varepsilon)\}\), which ends the proof of (a). Now, let \(0 < \varepsilon< \frac{1}{\beta}\). By item (a) we obtain
$$\begin{aligned} \bigl\vert F_{\varepsilon}(t) - \varepsilon^{-1}E(t)\bigr\vert = & \bigl\vert \varepsilon\Psi(t) - \Phi(t)\bigr\vert \\ \leq& \beta\bigl(E(t) + C_{\rho,\lambda_{1},h,\Omega}\bigr), \end{aligned}$$
which provides
$$\biggl(\frac{1}{\varepsilon} - \beta \biggr)E(t) - \frac{1}{2}C_{\rho ,\lambda_{1},h,\Omega} \leq F_{\varepsilon}(t) \leq \biggl(\frac {1}{\varepsilon} + \beta \biggr)E(t) + \frac{1}{2}C_{\rho,\lambda _{1},h,\Omega}. $$
This proves (b). □

Lemma 3.3

There exists \(\varepsilon_{1} > 0\) such that
$$F_{\varepsilon}'(t) \leq-\varepsilon E(t) + C_{\varepsilon}, \quad \forall t\geq0, \forall \varepsilon\in(0,\varepsilon_{1}). $$

Proof

By (3.3) we get
$$\begin{aligned} \Psi'(t) = & (u_{tt},u) + (u_{t},u_{t}) + (u_{tt},u)_{1} + (u_{t},u_{t})_{1} \\ = & \|u_{t}\|^{2} + \|u_{t}\|_{1}^{2} - \|u\|_{1}^{2} - (u,\eta)_{\mathcal{M}} -\bigl(f(u),u\bigr) + (h,u) \\ \leq& \|u_{t}\|^{2} + \|u_{t} \|_{1}^{2} - \|u\|_{1}^{2} - (u, \eta)_{\mathcal{M}} + \rho|\Omega| + \biggl( - \int_{\Omega} \hat{f}(u) \,dx + (h,u) \biggr) \\ = & \|u_{t}\|^{2} + \|u_{t}\|_{1}^{2} - \|u\|_{1}^{2} - (u,\eta)_{\mathcal{M}} + \rho|\Omega| - E(t) + \frac{1}{2}\|u_{t}\|^{2} + \frac{1}{2} \|u_{t}\|_{1}^{2} \\ &{} + \frac{1}{2}\|u\|_{1}^{2} + \frac{1}{2}\| \eta\|_{\mathcal{M}}^{2} \\ \leq& - E(t) + \frac{3}{2}\|u_{t}\|^{2} + \frac{3}{2}\|u_{t}\|_{1}^{2} - \frac {1}{2}\|u\|_{1}^{2} + \frac{1}{2}\|\eta \|_{\mathcal{M}}^{2} - (u,\eta )_{\mathcal{M}} + \rho|\Omega| \\ \leq& - E(t) + \bigl(1 + \lambda_{1}^{-1}\bigr) \|u_{t}\|_{1}^{2} - \frac{1}{2}\|u \|_{1}^{2} + \frac{1}{2}\|\eta\|_{\mathcal{M}}^{2} + \frac{1}{4}\|u\|_{1}^{2} + k_{0} \| \eta \|_{\mathcal{M}}^{2} + \rho|\Omega| \\ \leq& - E(t) + \bigl(1 + \lambda_{1}^{-1}\bigr) \|u_{t}\|_{1}^{2} - \frac{1}{4}\|u \|_{1}^{2} + \biggl(\frac{1}{2} + k_{0} \biggr)\|\eta\|_{\mathcal{M}}^{2} + \rho|\Omega| \end{aligned}$$
and
$$\begin{aligned} - \Phi'(t) = & - (u_{tt},\eta)_{\mathcal{M}} - (u_{t},\eta_{t})_{\mathcal {M}} - (u_{tt}, \eta)_{\mathcal{M}_{0}} - (u_{t},\eta_{t})_{\mathcal{M}_{0}} \\ = & - (u_{tt},\eta)_{\mathcal{M}} - (u_{tt}, \eta)_{\mathcal{M}_{0}} - \|u_{t}\|_{\mathcal{M}}^{2} - (T \eta,u_{t})_{\mathcal{M}} - \|u_{t}\|_{\mathcal {M}_{0}}^{2} + (T\eta,u_{t})_{\mathcal{M}_{0}} \\ = & - \|u_{t}\|_{\mathcal{M}}^{2} - \|u_{t} \|_{\mathcal{M}_{0}}^{2} - (T\eta, u_{t})_{\mathcal{M}} - (T \eta,u_{t})_{\mathcal{M}_{0}} + (u,\eta)_{\mathcal {M}} + \bigl(f(u),\eta \bigr)_{\mathcal{M}_{0}} \\ &{} - (h,\eta)_{\mathcal{M}_{0}} - \int_{0}^{\infty} \int_{0}^{\infty} \mu (s)\mu(\tau) \bigl(\eta(\tau), \eta(s)\bigr)_{1} \,d\tau \,ds \\ = & - \|u_{t}\|_{\mathcal{M}}^{2} - \|u_{t} \|_{\mathcal{M}_{0}}^{2} + \int _{0}^{\infty}\mu'(s) (\eta, u_{t})_{1}\,ds + \int_{0}^{\infty}\mu'(s) (\eta, u_{t})\,ds + \bigl(f(u),\eta\bigr)_{\mathcal{M}_{0}} \\ &{} - (h,\eta)_{\mathcal{M}_{0}} - \int_{0}^{\infty} \int_{0}^{\infty} \mu (s)\mu(\tau) \bigl(\eta(\tau), \eta(s)\bigr)_{1} \,d\tau \,ds \\ \leq& - \|u_{t}\|_{\mathcal{M}}^{2} - \|u_{t} \|_{\mathcal{M}_{0}}^{2} - k_{1}(\eta ,u_{t})_{\mathcal{M}} - k_{1}(\eta,u_{t})_{\mathcal{M}_{0}} + \bigl(f(u),\eta \bigr)_{\mathcal{M}_{0}} - (h,\eta)_{\mathcal{M}_{0}} \\ &{} - \int_{0}^{\infty} \int_{0}^{\infty} \mu(s)\mu(\tau) \bigl(\eta(\tau),\eta (s)\bigr)_{1} \,d\tau \,ds \\ \leq& - \frac{1}{2}\|u_{t}\|_{\mathcal{M}}^{2} - \frac{1}{2}\|u_{t}\| _{\mathcal{M}_{0}}^{2} + k_{1}^{2}\|\eta\|_{\mathcal{M}}^{2} + k_{1}^{2}\|\eta\| _{\mathcal{M}_{0}}^{2} + \bigl(f(u), \eta\bigr)_{\mathcal{M}_{0}} - (h,\eta)_{\mathcal {M}_{0}} \\ &{} - \int_{0}^{\infty} \int_{0}^{\infty} \mu(s)\mu(\tau) \bigl(\eta(\tau),\eta (s)\bigr)_{1} \,d\tau \,ds \\ \leq& -\frac{k_{0}}{2} \bigl(1+ \lambda_{1}^{-1} \bigr) \|u_{t}\|_{1} + k_{1}^{2}\bigl(1 + \lambda_{1}^{-1}\bigr) \|\eta\|_{\mathcal{M}}^{2} + \frac{\delta}{4}\|u\| _{1}^{2} + C_{\delta}\|\eta \|_{\mathcal{M}}^{2} \\ &{} + \frac{\lambda_{1}^{-1}k_{0}}{2}\|h\|^{2} + \frac{1}{2}\|\eta\| _{\mathcal{M}}^{2} , \end{aligned}$$
where \(0 < \delta\leq1\) and \(C_{\delta}\) is a positive constant that verifies the inequality
$$\begin{aligned} \bigl\vert \bigl(f(u),\eta\bigr)_{\mathcal{M}_{0}}\bigr\vert \leq & \int_{0}^{\infty}\mu(s) \int _{\Omega} \bigl(1 + |u|^{p}\bigr)|u||\eta| \,dx \,ds \\ \leq & \int_{0}^{\infty}\mu(s) \int_{\Omega} |u||\eta| \,dx \,ds + \int_{0}^{\infty}\mu(s) \int_{\Omega} |u|^{p+1}|\eta| \,dx \,ds \\ \leq & \lambda_{1}^{-1} \int_{0}^{\infty}\mu(s)\|u\|_{1}\|\eta \|_{1} \,ds + \int_{0}^{\infty}\mu(s) \|u\|_{p+2}^{p+1} \|\eta\|_{p+2} \,ds \\ \leq & \lambda_{1}^{-1} \int_{0}^{\infty}\mu(s)\|u\|_{1}\|\eta \|_{1} \,ds + K_{\Omega} \int_{0}^{\infty}\mu(s) \|u\|_{1}\|\eta \|_{1} \,ds \\ \leq &K_{\Omega}\|u\|_{1} \int_{0}^{\infty}\mu(s)\|\eta\|_{1} \,ds \\ \leq & \frac{\delta}{4} \|u\|_{1}^{2} + C_{\delta} \|\eta\|_{\mathcal{M}}^{2}. \end{aligned}$$
Now, for every \(0<\varepsilon<\frac{k_{0}}{2}\), the above inequalities provide
$$\begin{aligned} F_{\varepsilon}'(t) - \varepsilon^{-1}E'(t) = & \varepsilon\Psi'(t) - \Phi'(t) \\ \leq& \bigl(1 + \lambda_{1}^{-1}\bigr) \biggl(\varepsilon- \frac{k_{0}}{2} \biggr)\| u_{t}\|_{1} -\varepsilon E(t) + \frac{1}{4}(\delta- \varepsilon)\|u\|_{1}^{2} + \tilde{C}_{\varepsilon,\delta} \\ \leq& -\varepsilon E(t) + \frac{1}{4}(\delta- \varepsilon)\|u\| _{1}^{2} + \tilde{C}_{\varepsilon,\delta}, \end{aligned}$$
where \(\tilde{C}_{\varepsilon,\delta}\) is a positive constant. As \(E'(t) \leq0\) we can choose \(\delta\leq\varepsilon\) in the previous inequality to obtain
$$F_{\varepsilon}'(t) \leq- \varepsilon E(t) + C_{\varepsilon}, $$
which ends the proof of the lemma. □

Lemma 3.4

(Absorbing set)

Let \(S(t)\) be the \(C_{0}\)-semigroup defined in (2.5). Then \((\mathcal{H},S(t))\) is a dissipative dynamical system.

Proof

We shall prove that \(S(t)\) has a bounded absorbing set. Let \(\varepsilon_{0}=\min\{\frac{1}{2\beta},\varepsilon_{1}\}\). By item (b) of Lemma 3.2 we have
$$ \beta_{1} E(t) - \frac{1}{2} C_{\rho,\lambda_{1},h,\Omega} \leq F_{\varepsilon}(t) \leq\beta_{2} E(t) + \frac{1}{2} C_{\rho,\lambda _{1},h,\Omega}, $$
(3.7)
where \(\beta_{1}=\varepsilon^{-1} - \beta\) and \(\beta_{2}=\varepsilon^{-1} + \beta\). Multiplying the inequality (3.7) for \(\frac{\varepsilon}{\beta_{2}}\) we get
$$ \frac{\varepsilon}{\beta_{2}}F_{\varepsilon}(t) \leq\varepsilon E(t) + \mathbf{a}_{\varepsilon} , $$
(3.8)
where \(\mathbf{a}_{\varepsilon}=\varepsilon\beta_{2}^{-1}C_{\rho ,\lambda_{1},h,\Omega}\). Now by Lemma 3.3 we have
$$ F_{\varepsilon}'(t) \leq- \varepsilon E(t) + C_{\varepsilon}. $$
(3.9)
Adding the inequalities (3.8) and (3.9) we obtain
$$ F_{\varepsilon}'(t) + \frac{\varepsilon}{\beta_{2}}F_{\varepsilon}(t) \leq \mathbf{b}_{\varepsilon}, $$
(3.10)
where \(\mathbf{b}_{\varepsilon} = \mathbf{a}_{\varepsilon}+ C_{\varepsilon}\). By (3.10) we conclude that
$$ F_{\varepsilon}(t) \leq \bigl(F_{\varepsilon}(0) - \varepsilon ^{-1} \mathbf{b}_{\varepsilon}\beta_{2} \bigr)e^{-\frac{\varepsilon }{\beta_{2}}t} + \varepsilon^{-1}\mathbf{b}_{\varepsilon}\beta_{2},\quad \mbox{for all } t \geq0. $$
(3.11)
But by inequality (3.8) we have \(F_{\varepsilon}(0) \leq \beta_{2} E(0) + \varepsilon^{-1}\mathbf{a}_{\varepsilon}\beta_{2}\). Therefore, by inequality (3.11) we get
$$ F_{\varepsilon}(t) \leq \bigl(\beta_{2} E(0) - \varepsilon ^{-1}C_{\varepsilon}\beta_{2} \bigr)e^{-\frac{\varepsilon}{\beta_{2}}t} + \varepsilon^{-1}\mathbf{b}_{\varepsilon}\beta_{2}, \quad \mbox{for all } t \geq0. $$
(3.12)
Combining (3.7) and (3.12) we obtain
$$\begin{aligned} E(t) \leq & \frac{\beta_{2}}{\beta_{1}}\bigl(E(0) - \varepsilon^{-1} C_{\varepsilon}\bigr)e^{-\frac{\varepsilon}{\beta_{2}}t} + \frac{\beta_{2}}{\beta _{1}}C_{\varepsilon} + \frac{3}{2\beta_{1}}C_{\rho,\lambda_{1},h,\Omega} \\ \leq & \frac{\beta_{2}}{\beta_{1}}E(0)e^{-\frac{\varepsilon}{\beta_{2}}t} + \frac{\beta_{2}}{\beta_{1}}C_{\varepsilon} + \frac{3}{2\beta_{1}}C_{\rho ,\lambda_{1},h,\Omega}. \end{aligned}$$
(3.13)
By (3.6) we have
$$ E(t) \geq\sigma\bigl\Vert \bigl(u(t),u_{t}(t),\eta^{t} \bigr)\bigr\Vert _{\mathcal{H}}^{2} - C_{\rho ,\lambda_{1},h,\Omega}, \quad \mbox{for all } t\geq0, $$
(3.14)
where \(\sigma=\min\{1,\frac{\lambda_{1}}{2}\}\). Combining (3.13) and (3.14) we get
$$ \bigl\Vert \bigl(u(t),u_{t}(t),\eta^{t}\bigr)\bigr\Vert _{\mathcal{H}}^{2} \leq\frac{\beta_{2}}{\beta _{1}\sigma}E(0)e^{-\frac{\varepsilon}{\beta_{2}}t} + \frac{\beta_{2}}{\beta _{1}\sigma}C_{\varepsilon} + \biggl(\frac{3}{2\beta_{1}\sigma} + \frac {1}{\sigma} \biggr)C_{\rho,\lambda_{1},h,\Omega}. $$
(3.15)
By inequality (3.15) the ball \(B(0,R) \subset\mathcal {H}\), where
$$R>\sqrt{\frac{2\beta_{2}}{\beta_{1}\sigma}C_{\varepsilon} + \biggl(\frac {3}{\beta_{1}\sigma} + \frac{2}{\sigma} \biggr)C_{\rho,\lambda_{1},h,\Omega}}, $$
is an absorbing set of the semigroup \(S(t)\). □

3.3 Compactness

In this section we shall prove that the system \((\mathcal{H},S(t))\) is asymptotically smooth.

Lemma 3.5

(Stabilization inequality)

Let \({B}\subset\mathcal{H}\) be a bounded invariant set and \(z=(u_{0},v_{0},\eta_{0})\), \(\tilde{z}=(\tilde{u}_{0},\tilde{v}_{0},\tilde{\eta }_{0})\) two initial data in B. Then there exists \(\nu> 0\) such that
$$ \bigl\Vert S(t)z - S(t) \tilde{z} \bigr\Vert _{\mathcal{H}}^{2} \le C_{B} e^{-\nu t} + C_{B} \int_{0}^{t} \bigl(\bigl\Vert w(s) \bigr\Vert _{L^{p+2}}^{2} + \bigl\Vert w_{t}(s)\bigr\Vert _{L^{p+2}}^{2} \bigr) \,ds, $$
(3.16)
where \((u,\eta)\), \((\tilde{u}, \tilde{\eta})\) are the corresponding weak solutions of (2.1)-(2.2), \(w=u-\tilde{u}\), and \(C_{B}\) is a positive constant depending on B but not on t.

Proof

Let us also write \(\xi=\eta- \tilde{\eta}\). Then w is a weak solution of the system
$$ \left \{ \textstyle\begin{array}{l} w_{tt} - \Delta w_{tt} - \Delta w - \int_{0}^{\infty}\mu(s) \Delta\xi(s) \,ds = f(u) - f(\tilde{u}), \\ \xi_{t} = T \xi+ w_{t} , \end{array}\displaystyle \right . $$
(3.17)
with Dirichlet boundary condition and initial data
$$w(0)=u_{0} - \tilde{u}_{0}, \qquad w_{t}(0)=v_{0} - \tilde{v}_{0}, \qquad \xi^{0} =\eta_{0} = \tilde{ \eta}_{0}. $$
We define the energy functional
$$G(t) = \frac{1}{2}\bigl\Vert w_{t}(t)\bigr\Vert ^{2} + \frac{1}{2}\bigl\Vert w_{t}(t)\bigr\Vert _{1}^{2} + \frac {1}{2}\bigl\Vert w(t)\bigr\Vert _{1}^{2} + \frac{1}{2}\bigl\Vert \xi^{t} \bigr\Vert _{\mathcal{M}}^{2} .$$
In the following, \(C_{0}\) will denote several positive constants dependent on B but not on t.

Claim 1

There exists a constant \(C_{0} >0\) such that
$$ G'(t) \le\frac{1}{2} \int_{0}^{\infty} \mu'(s) \bigl\Vert \xi^{t} (s)\bigr\Vert _{1}^{2} \,ds + C_{0} \bigl( \bigl\Vert w(t) \bigr\Vert _{L^{p+2}}^{2} + \bigl\Vert w_{t}(t) \bigr\Vert _{L^{p+2}}^{2} \bigr). $$
(3.18)
To prove the claim, we multiply the first equation in (3.17) by \(w_{t}\) and integrate over Ω. Then we obtain
$$G'(t)= \frac{1}{2} \int_{0}^{\infty} \mu'(s) \bigl\Vert { \xi}(s) \bigr\Vert _{1}^{2} \,ds - \int_{\Omega} \bigl( f(u) - f(\tilde{u}) \bigr) w_{t} \, dx . $$
Using (H2) we have
$$\begin{aligned} \biggl\vert \int_{\Omega} \bigl( f(u) - f(\tilde{u}) \bigr) w_{t}\, dx \biggr\vert \le& C_{f} \bigl( 1 + \Vert u \Vert_{L^{p+2}}^{p} + \Vert\tilde{u} \Vert_{L^{p+2}}^{p} \bigr) \Vert w \Vert_{L^{p+2}} \Vert w_{t} \Vert _{L^{p+2}} \\ \le& C_{0} \bigl( \bigl\Vert w(t) \bigr\Vert _{L^{p+2}}^{2} + \bigl\Vert w_{t}(t) \bigr\Vert _{L^{p+2}}^{2} \bigr), \end{aligned}$$
since B is bounded and invariant. Then we see that (3.18) holds.
Now, let us define the perturbed functional
$${J}(t) = N G(t) + \varepsilon\tilde{\Psi}(t) + \tilde{\Phi}(t) , $$
where
$$\tilde{\Psi}(t) = \bigl(w(t),w_{t}(t)\bigr) + \bigl(w(t),w_{t}(t) \bigr)_{1}, \qquad \tilde{\Phi }(t)= \bigl(w(t),\xi^{t} \bigr)_{\mathcal{M}} + \bigl(w(t),\xi^{t}\bigr)_{\mathcal{M}_{0}}, $$
and \(N \ge1\), \(0<\varepsilon<1\) are constants to be determined. Then the following claims can be proved with similar arguments to the above one and to the proof of the absorbing set.

Claim 2

There exist constants \(\beta_{1},\beta_{2},C_{\beta }>0\) such that, if \(N> C_{\beta}\),
$$ \beta_{1} G(t) \le J(t) \le\beta_{2} G(t), \quad t \ge0. $$
(3.19)

Claim 3

There exists a constant \(C_{1}>0\) such that
$$\begin{aligned} \tilde{\Psi}'(t) \le& -G(t) - \frac{1}{4} \bigl\Vert w(t) \bigr\Vert _{1}^{2} + \frac{3}{2} \bigl\Vert w_{t}(t) \bigr\Vert ^{2} + \frac{3}{2} \bigl\Vert w_{t}(t) \bigr\Vert _{1}^{2} \\ &{} - C_{1} \int_{0}^{\infty} \mu'(s) \bigl\Vert \xi^{t} (s)\bigr\Vert _{1}^{2} \,ds + C_{0} \bigl\Vert w(t) \bigr\Vert _{L^{p+2}}^{2} . \end{aligned}$$
(3.20)

Claim 4

Given \(\delta>0\) there exists a constant \(C_{\delta} >0\) such that
$$ \tilde{\Phi}'(t) \le\delta\bigl\Vert w(t) \bigr\Vert _{1}^{2} - \frac{k_{0}}{2} \bigl\Vert w_{t}(t) \bigr\Vert _{1}^{2} - C_{\delta} \int_{0}^{t} \mu'(s) \bigl\Vert \xi^{t} (s)\bigr\Vert _{1}^{2} \,ds. $$
(3.21)
Now, taking \(\varepsilon>0\) sufficiently small and \(N>0\) sufficiently large, we obtain from (3.18), (3.20), and (3.21),
$$J'(t) \le- \varepsilon G(t) + C_{0} \bigl( \bigl\Vert w(t) \bigr\Vert _{L^{p+2}}^{2} + \bigl\Vert w_{t}(t) \bigr\Vert _{L^{p+2}}^{2} \bigr) , \quad t \ge0. $$
Combining this with (3.19) we have, as in the proof of the absorbing set,
$$G (t) \le\frac{\beta_{2}}{\beta_{1}} G(0) e^{-\frac{\varepsilon}{\beta_{2}}t} + C_{0} \int_{0}^{t} e^{- \frac{\varepsilon}{\beta_{2}}(t-s)} \bigl( \bigl\Vert w(s) \bigr\Vert _{L^{p+2}}^{2} + \bigl\Vert w_{t}(s) \bigr\Vert _{L^{p+2}}^{2} \bigr) \,ds,\quad \forall t \ge0. $$
This implies (3.16) by taking \(\nu= \varepsilon/ \beta_{2}\) and in view of the definition of \(G(t)\).  □

Lemma 3.6

(Asymptotic smoothness)

Let \(S(t)\) be the \(C_{0}\)-semigroup defined in (2.5). Then the system \((\mathcal{H},S(t))\) is asymptotically smooth.

Proof

We apply the compactness criterion presented in Proposition 2.10 of [18]. As recalled in Section 3.1, we must check conditions (3.1) and (3.2).

Given a forward invariant set \(B \subset\mathcal{H}\) and \(\varepsilon >0\), we can take \(T>0\) such that
$$\sqrt{2C_{B}} e^{-\frac{\nu}{2} T} < \varepsilon. $$
Then from (3.16), using notation
$$S(t)z^{n} = \bigl(u^{n}(t),u^{n}_{t}(t), \eta_{n}^{t}\bigr), $$
we obtain for any \(z^{1},z^{2} \in B\),
$$ \bigl\Vert S(t)z^{1} - S(t)z^{2} \bigr\Vert _{\mathcal{H}} \le\varepsilon+ \biggl( 2C_{B} \int_{0}^{T} \bigl\Vert u^{1}- u^{2} \bigr\Vert _{L^{p+2}}^{2} + \bigl\Vert u^{1}_{t} - u^{2}_{t} \bigr\Vert _{L^{p+2}}^{2} \,ds \biggr)^{\frac{1}{2}} $$
with \(0< t< T\). Then defining
$$\phi_{T}\bigl(z^{1},z^{2}\bigr)= \sqrt{2C_{B}} \biggl( \int_{0}^{T} \bigl\Vert u^{1}(s)- u^{2}(s) \bigr\Vert _{L^{p+2}}^{2} + \bigl\Vert u^{1}_{t}(s) - u^{2}_{t}(s) \bigr\Vert _{L^{p+2}}^{2} \,ds \biggr)^{\frac{1}{2}}, $$
we see that condition (3.1) holds.
It remains to show that (3.2) also holds. Given any sequence \((z^{n}) \subset B\), from the positive invariance of B we see that \(S(t)z^{n} = (u^{n}(t),u^{n}_{t}(t),\eta_{n}^{t})\) is uniformly bounded in \(\mathcal{H}\). Then we conclude that
$$\begin{aligned}& u^{n} \quad \mbox{is bounded in } L^{\infty}(0,T,H_{1}), \\& u^{n}_{t} \quad \mbox{is bounded in } L^{\infty} \bigl(0,T,L^{2}(\Omega)\bigr) \cap L^{\infty}(0,T,H_{1}) , \end{aligned}$$
and from (2.4),
$$u^{n}_{tt} \quad \mbox{is bounded in } L^{2} \bigl(0,T,L^{2}(\Omega)\bigr). $$
Then from Simon’s theorem [22] we have
$$u^{n}, u^{n}_{t}\quad \mbox{converge strongly in } C\bigl([0,T],L^{p+2}(\Omega)\bigr) , $$
since \(H_{1}\) is compactly embedded in \(L^{p+2}(\Omega)\). Therefore there is a subsequence such that
$$\lim_{k\to\infty} \lim_{l\to\infty} \int_{0}^{T} \bigl\Vert u^{n_{k}}(s)- u^{n_{l}}(s) \bigr\Vert _{L^{p+2}}^{2} + \bigl\Vert u^{n_{k}}_{t}(s) - u^{n_{l}}_{t}(s) \bigr\Vert _{L^{p+2}}^{2} \,ds = 0. $$
This shows that (3.2) also holds. □

Proof of Theorem 3.1

Since we have proved that \((\mathcal{H},S(t))\) is dissipative and asymptotically smooth, the existence of a global attractor follows from a classical result, as noticed in Section 3.1. □

Declarations

Acknowledgements

This work was supported by the National Natural Science Foundation of China with grant number 11526164.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
College of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, P.R. China
(2)
Department of Sciences, Regional Campus of Goioerê, State University of Maringá, Goioerê, Paraná, 87360-000, Brazil
(3)
Department of Mathematics, State University of Maringá, Maringá, Paraná, 87020-900, Brazil

References

  1. Love, AEH: Treatise on Mathematical Theory of Elasticity. Dover, New York (1944) MATHGoogle Scholar
  2. Cavalcanti, MM, Domingos Cavalcanti, VN, Ferreira, J: Existence and uniform decay for a nonlinear viscoelastic equation with strong damping. Math. Methods Appl. Sci. 24, 1043-1053 (2001) View ArticleMathSciNetMATHGoogle Scholar
  3. Han, X, Wang, M: Global existence and uniform decay for a nonlinear viscoelastic equation with damping. Nonlinear Anal. 70, 3090-3098 (2009) View ArticleMathSciNetMATHGoogle Scholar
  4. Messaoudi, SA, Tatar, N-E: Exponential decay for a quasilinear viscoelastic equation. Math. Nachr. 282, 1443-1450 (2009) View ArticleMathSciNetMATHGoogle Scholar
  5. Wu, S-T: General decay of solutions for a nonlinear system of viscoelastic wave equations with degenerate damping and source terms. J. Math. Anal. Appl. 406, 34-48 (2013) View ArticleMathSciNetMATHGoogle Scholar
  6. Araújo, RO, Ma, TF, Qin, Y: Long-time behavior of a quasilinear viscoelastic equation with past history. J. Differ. Equ. 254, 4066-4087 (2013) View ArticleMATHGoogle Scholar
  7. Qin, Y, Feng, B, Zhang, M: Uniform attractors for a non-autonomous viscoelastic equation with a past history. Nonlinear Anal. 101, 1-15 (2014) View ArticleMathSciNetMATHGoogle Scholar
  8. Chen, C, Wang, H, Zhu, S: Global attractor and decay estimates of solutions to a class of nonlinear evolution equations. Math. Methods Appl. Sci. 34, 497-508 (2011) View ArticleMathSciNetMATHGoogle Scholar
  9. Conti, M, Marchini, EM, Patta, V: A well posedness result for nonlinear viscoelastic equations with memory. Nonlinear Anal. 94, 206-216 (2014) View ArticleMathSciNetMATHGoogle Scholar
  10. Cavalcanti, MM, Fatori, LH, Ma, TF: Attractors for wave equations with degenerate memory. J. Differ. Equ. 260, 56-83 (2016) View ArticleMathSciNetGoogle Scholar
  11. Gatti, S, Miranville, A, Pata, V, Zelik, S: Attractors for semi-linear equations of viscoelasticity with very low dissipation. Rocky Mt. J. Math. 38(4), 1117-1138 (2008) View ArticleMathSciNetMATHGoogle Scholar
  12. Giorgi, C, Rivera, JE, Pata, V: Global attractors for a semilinear hyperbolic equation in viscoelasticity. J. Math. Anal. Appl. 260, 83-99 (2001) View ArticleMathSciNetMATHGoogle Scholar
  13. Messaoudi, SA: General decay of solutions of a viscoelastic equation. J. Math. Anal. Appl. 341, 1457-1467 (2008) View ArticleMathSciNetMATHGoogle Scholar
  14. Pata, V, Zucchi, A: Attractors for a damped hyperbolic equation with linear memory. Adv. Math. Sci. Appl. 11, 505-529 (2001) MathSciNetMATHGoogle Scholar
  15. Lions, JL: Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires. Dunod, Paris (1969) MATHGoogle Scholar
  16. Dafermos, CM: Asymptotic stability in viscoelasticity. Arch. Ration. Mech. Anal. 37, 297-308 (1970) View ArticleMathSciNetMATHGoogle Scholar
  17. Babin, AV, Vishik, MI: Attractors of Evolution Equations. Studies in Mathematics and Its Applications, vol. 25. North-Holland, Amsterdam (1992) View ArticleMATHGoogle Scholar
  18. Chueshov, I, Lasiecka, I: Long-time behavior of second order evolution equations with nonlinear damping. Mem. Am. Math. Soc. 195, 912 (2008) MathSciNetGoogle Scholar
  19. Hale, JK: Asymptotic Behaviour of Dissipative Systems. Mathematical Surveys and Monographs, vol. 25. Am. Math. Soc., Providence (1988) Google Scholar
  20. Ladyzhenskaya, O: Attractors for Semi-Groups and Evolution Equations. Cambridge University Press, Cambridge (1991) View ArticleGoogle Scholar
  21. Temam, R: Infinite Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1997) View ArticleMATHGoogle Scholar
  22. Simon, J: Compact sets in the space \(L^{p}(0, T ; B)\). Ann. Mat. Pura Appl. 146, 65-96 (1987) View ArticleMathSciNetMATHGoogle Scholar

Copyright

© Feng et al. 2016

Advertisement