Research Article | Open Access
Stochastic Viscoelastic Wave Equations with Nonlinear Damping and Source Terms
The goal of this paper is to study an initial boundary value problem of stochastic viscoelastic wave equation with nonlinear damping and source terms. Under certain conditions on the initial data: the relaxation function, the indices of nonlinear damping, and source terms and the random force, we prove the local existence and uniqueness of solution by the Galerkin approximation method. Then, considering the relationship between the indices of nonlinear damping and nonlinear source, we give the necessary conditions of global existence and explosion in finite time in some sense of solutions, respectively.
We consider a stochastic viscoelastic wave equation with nonlinear damping and source terms where is a bounded domain in with smooth boundary , , , is a given positive constant which measures the strength of noise, and is an infinite dimensional Wiener process, is -valued progressively measurable and is a positive relaxation function satisfying some conditions to be specified later. By simplicity, we have set equal to 1 all the coefficients in the equation different from the random force.
For the deterministic case on viscoelastic wave equation, many authors studied the following problem: where . If , . Haraux and Zuazua  and Kopáčková  proved that the damping term assures existence of global solution and decay of solution for arbitrary initial data. If , , Ball  and Kalantarov and Ladyzhenskaya  gave that the source term causes finite time blow-up with the large initial data. If , , the interaction between damping term and source term occurs; Levine et al. [5, 6] studied the linear damping (i.e., ) and proved that the solution with negative initial energy blows up in finite time; Georgiev and Todorova  considered nonlinear damping and source terms; they showed that the solution blows up in finite time if for sufficiently large initial data and exists globally if with large initial data. Alves et al.  and Rammaha  focused the nonlinear wave equations or systems on the influence between damping and source and described the existence, uniform decay rates, and blow-up to the solutions.
In fact, lots of investigators have paid attention to the viscoelastic wave equation, which has its origin in the mathematical description of viscoelastic materials. The dynamic properties of viscoelastic materials are of great importance as they appear in many applications to natural sciences. The general viscoelastic wave equation has the following form: where is a relaxation function and , are given functions. If is nonlinear, is linear; Kafini and Messaoudi  established a blow-up result; if , . Messaoudi  showed that the solution with negative initial energy blows up in finite time if and exists globally if under suitable conditions on relaxation function . This blow-up result has been pushed to the case of positive initial energy by Messaoudi . For , Song and Zhong  obtained that the solution with positive initial energy blows up in finite time. Ikehata  gave some remarks on the wave equations with nonlinear damping and source terms. Aassila et al. , Cavalcanti et al. , and Cavalcanti et al.  studied the boundary damping and proved the existence and uniform decay of the solutions. Cavalcanti et al.  discussed the asymptotic stability of the wave equation on a compact Riemannian manifold; they proved that the solutions of the corresponding partial viscoelastic model decay exponentially to zero under some conditions.
Under the consideration of random environment, some authors investigated the following stochastic wave equation with nonlinear damping and source terms: where is a -valued -Wiener process on some completed probability space, and is a nonnegative operator with finite trace on (see [19–26]). If , or ; Bo et al.  showed that the solution blows up with positive probability or it is explosive in sense. If , Gao et al.  showed that the global solution exists for , and the solution blows up with positive probability or is explosive in energy sense for .
Recently, Wei and Jiang  and Liang and Gao  considered the following nonlinear stochastic viscoelastic wave equation with linear damping: and they used the fixed point theorem to prove the existence and uniqueness of local mild solution; then by an appropriate energy inequality and estimations, they obtained the global existence and the decay estimate of the energy function of the solution and showed that the solution blows up with positive probability or it is explosive in sense under some conditions.
As we know, no one considers the stochastic viscoelastic wave equation (1) with the interaction between nonlinear damping and nonlinear source terms. In this paper, we study the global existence and the explosive phenomena under some suitable conditions on the nonlinear damping and nonlinear source terms.
In contrast with the model in , we add a viscoelastic term and use the nonlinear damping term instead of the linear damping and the strong damping . To the model in , we add a viscoelastic term . To the model in [29, 30], we use the nonlinear damping term instead of the linear damping . To the model in , we add a random force. In this paper, we generalize the blow-up and global existence results to the solution of (1) with interaction among viscoelastic memory, nonlinear damping, nonlinear source, and random force.
This paper is organized as follows. In the next section, we recall some preliminaries related to assumptions and definitions for the solutions of the stochastic equations. In Section 3, we use the Galerkin approximation method to get the local solution of stochastic viscoelastic wave equations with nonlinear damping and source terms. In Section 4, by the energy function and some estimates, we prove that the solution blows up with positive probability or it is explosive in energy sense for . In the last section, we obtain the existence of global solution by the Borel-Cantelli Lemma.
Let be a separable Hilbert space with Borel -algebra , and let be a probability space. We set with the inner product and norm denoted by and , respectively. Denote by the norm for and by the Dirichlet norm in which is equivalent to norm. We also assume that , satisfy
Lemma 1 (see ). For all and or , there exists a constant such that
One assumes that is a bounded nonincreasing function satisfying , , and there exist positive constants and such that
In this paper, stands for expectation with respect to probability measure , is a -valued -Wiener process on the probability space with the covariance operator satisfying . A complete orthonormal system in with , and a bounded sequence of nonnegative real numbers satisfies that , .
To simplify the computations, we assume that the covariance operator and Laplacian with homogeneous Dirichlet boundary condition have a common set of eigenfunctions, that is, and then, for any , has an expansion where are real valued Brownian motions mutually independent on . Let be the set of -valued processes with the norm where denotes the adjoint operator of . For any process , we can define the stochastic integral with respect to the -Wiener process as , which is a martingale. For more details about the infinite dimension Wiener process and the stochastic integral, we refer the readers to .
Definition 2. Assume that , and ; is said to be a solution of (1) on the interval , if is -valued progressively measurable, , , and such that (1) holds in the sense of distributions over for almost all .
3. Local Existence and Uniqueness
In this section, we establish the local existence and uniqueness of solution to problem (1) by the Galerkin approximation method. Set , . For each , we define a cut-off function , such that , for , and
Denote for , then, Lemma 1 implies that where is a constant depending only on .
Lemma 3 (see ). Let be a sequence of positive numbers, and let be a sequence of real numbers such that and as , then
Lemma 4 (see ). Let be a bounded domain in . Suppose that is a bounded sequence in , such that for almost all , for some . Then weakly in .
Fix and ; we consider the regularized initial boundary value problem with the initial data and is -valued progressively measurable such that
For notational convenience, we omit in the Hilbert space.
Proof. Let and let be the solution of the following system:
By formula, we have
for all and almost all , where
Using Hölder’s inequality, Young’s inequality, Poincaré’s inequality, and Lemma 1, we have
and so we have
Since thus we have where depends on the fixed number .
Due to integration by parts, we have Since is the Yosida approximation of mapping , by the Lemma 3 and sign-preserving theorem of limit, for small enough , we have Moreover, by the conditions of , we get Next, from the properties of , we have and similar to the derivation of (34), we have By the B-D-G inequality and Young’s inequality, we have
From (22)–(37), we get then, Set and we can rewrite (39) as follows: and due to Gronwall’s inequality and (29), it is clear that
Let be the orthogonal projection of into the space Span , such that Define , where is an -valued progressively measurable such that (19) holds and is an -valued process; there is a subset with such that for each , , and we have for all .
From (42), there is a subsequence , for each , such that, and by the properties of relaxation function and Hölder’s inequality, we get From (15) and embedding theorem we have Together with (42)–(47), we obtain for all . By (48) and weakly star in , we have This implies that there exists a subsequence still denoted by such that Due to (47) and Lemma 4, it is clear that Thus, satisfies (17) in the sense of distributions over .
Next, we will prove the uniqueness of the solution. If there is another solution of (17), , in the above sense, then satisfies Taking the inner product of (51) with in , we obtain From (15), we have By Lemma 1 and Hlder’s inequality, Due to (30), we have Combining (52) with (55), similar to (32) and (34), we get which implies , that is, . So is well defined, for each .
Finally, we state that is -valued progressively measurable for any , and the energy inequality holds true; this can be established by the similar argument in [28, 31].
Moreover, we still fix and consider the following problem: The following lemma is important to prove the local existence of solution of (1).
Lemma 6 (see [28, 31]). Assume that (18), (19), and the conditions of hold. Then there is a pathwise unique solution of (58) such that From Lemmas 5 and 6, we state a local existence theorem of (1); the proof is standard; for more information we refer the readers to [28, 31].
4. Blow Up
In this section we prove our main result for . For this purpose, we give refined restrictions on and relaxation function such that Define an energy function where
For each , introduce the stopping time by , where is increasing in , let .
In order to prove our blow-up result, we rewrite (1) as an equivalent system where . Then the energy function becomes
First we give a lemma.
Proof. Using formula to and , respectively, and taking the expectations, in the same way as our discussions in existence of solution to deal with the memory term, it is easy to get (66) and (67) (see ).
Let Due to (61), we have