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Long-time behavior of a semilinear wave equation with memory
Boundary Value Problems volume 2016, Article number: 37 (2016)
Abstract
In this paper we study the long-time dynamics of the semilinear viscoelastic equation
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
with initial conditions
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
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
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
is self-adjoint and strictly positive. See, e.g., [15]. We adopt the notation
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
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
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,
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
We also introduce the Hilbert space
endowed with the norms \(\|\cdot\|_{\mathcal{H}}=(\cdot,\cdot)_{\mathcal{H}}^{\frac{1}{2}}\) where
Then, as in [12, 14], we define
Using this new variable η we can reformulate the system (1.1)-(1.3) to become
with initial conditions
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
satisfying the identity
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 \(\mathcal{M}\),
where τ is a positive real number arbitrarily fixed. In addition, it was shown in Estimate (4.4) of [9],
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
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,
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 \(s \in\mathbb{R}^{+}\).
-
(i)
- (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 \(\hat{f}(s) = \int_{0}^{s} f(\tau)\, d\tau\).
-
(i)
- (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
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
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
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
where \(\phi_{T}: {B} \times{B} \to\mathbb{R}\) satisfies
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
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
We are going to apply the perturbed energy method. We consider the maps
and
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
where \(|\Omega|\) is the measure of Ω. Therefore, there is a positive constant \(C_{\rho,\lambda_{1},h,\Omega}\) such that
Applying the inequality (3.6) we obtain
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
which provides
This proves (b). □
Lemma 3.3
There exists \(\varepsilon_{1} > 0\) such that
Proof
By (3.3) we get
and
where \(0 < \delta\leq1\) and \(C_{\delta}\) is a positive constant that verifies the inequality
Now, for every \(0<\varepsilon<\frac{k_{0}}{2}\), the above inequalities provide
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
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
where \(\beta_{1}=\varepsilon^{-1} - \beta\) and \(\beta_{2}=\varepsilon^{-1} + \beta\). Multiplying the inequality (3.7) for \(\frac{\varepsilon}{\beta_{2}}\) we get
where \(\mathbf{a}_{\varepsilon}=\varepsilon\beta_{2}^{-1}C_{\rho ,\lambda_{1},h,\Omega}\). Now by Lemma 3.3 we have
Adding the inequalities (3.8) and (3.9) we obtain
where \(\mathbf{b}_{\varepsilon} = \mathbf{a}_{\varepsilon}+ C_{\varepsilon}\). By (3.10) we conclude that
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
Combining (3.7) and (3.12) we obtain
By (3.6) we have
where \(\sigma=\min\{1,\frac{\lambda_{1}}{2}\}\). Combining (3.13) and (3.14) we get
By inequality (3.15) the ball \(B(0,R) \subset\mathcal {H}\), where
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
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
with Dirichlet boundary condition and initial data
We define the energy functional
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
To prove the claim, we multiply the first equation in (3.17) by \(w_{t}\) and integrate over Ω. Then we obtain
Using (H2) we have
since B is bounded and invariant. Then we see that (3.18) holds.
Now, let us define the perturbed functional
where
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 \(N> C_{\beta}\),
Claim 3
There exists a constant \(C_{1}>0\) such that
Claim 4
Given \(\delta>0\) there exists a constant \(C_{\delta} >0\) such that
Now, taking \(\varepsilon>0\) sufficiently small and \(N>0\) sufficiently large, we obtain from (3.18), (3.20), and (3.21),
Combining this with (3.19) we have, as in the proof of the absorbing set,
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
Then from (3.16), using notation
we obtain for any \(z^{1},z^{2} \in B\),
with \(0< t< T\). Then defining
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
and from (2.4),
Then from Simon’s theorem [22] we have
since \(H_{1}\) is compactly embedded in \(L^{p+2}(\Omega)\). Therefore there is a subsequence such that
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. □
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This work was supported by the National Natural Science Foundation of China with grant number 11526164.
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Feng, B., Pelicer, M.L. & Andrade, D. Long-time behavior of a semilinear wave equation with memory. Bound Value Probl 2016, 37 (2016). https://doi.org/10.1186/s13661-016-0551-5
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DOI: https://doi.org/10.1186/s13661-016-0551-5