Long-time behavior of a semilinear wave equation with memory

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.

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., [2–5]. 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 [6–9]. 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 [10–14], 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:

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

• (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:

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

• (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., [17–21]. 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

• (a) there exist constants $\beta=\beta(k_{0},\lambda_{1}) \geq1$ and $C_{\rho,\lambda_{1},h,\Omega} \geq0$ such that

  $$\bigl\vert \varepsilon\Psi(t) - \Phi(t)\bigr\vert \leq\beta\bigl(E(t) + C_{\rho,\lambda _{1},h,\Omega}\bigr); $$

• (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. □