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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 403528, 12 pages
Stochastic Nonlinear Thermoelastic System Coupled Sine-Gordon Equation Driven by Jump Noise
1School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China
2School of Mathematics and Statistics, Xuchang University, Xuchang, Henan 46100, China
Received 23 November 2013; Accepted 13 January 2014; Published 24 February 2014
Academic Editor: Ming Mei
Copyright © 2014 Shuilin Cheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper considers a stochastic nonlinear thermoelastic system coupled sine-Gordon equation driven by jump noise. We first prove the existence and uniqueness of strong probabilistic solution of an initial-boundary value problem with homogeneous Dirichlet boundary conditions. Then we give an asymptotic behavior of the solution.
In this paper, we consider the following stochastic nonlinear thermoelastic coupled sine-Gordon system driven by Lévy noise: where , are the Lévy processes defined on a complete probability space (see Section 2 for the precise definition) [1–4], and are given real-valued random functions that will be defined in later.
Recently, the study of high-temperature apparatus or heat resistant structures is becoming important and it is necessary to analyze not only the deterministic thermal stress but also the stochastic thermal stress. In high-temperature apparatus, it is very difficult to predict accurately the thermal environment and mechanical load on its components. Furthermore, many indeterminate factors must be considered, for example, the random high-cycle vibrations of the temperature of the upper core in fast breeder reactors and fluctuations in the heat transfer coefficients around the stationary blades of gas turbines. Therefore, the stochastic case of temperature and thermal stress is indispensable in considering these indeterminate factors of the thermal environment (see ).
All the time, a description of wave propagation phenomena in random media is usually based on the study of stochastically perturbed wave equations (see [6, 7]). In fact, lots of wave phenomena are temperature dependent or heat generating; then the wave equations are coupled with a stochastic heat equation. Caraballo et al.  studied the existence of invariant manifolds for coupled parabolic and hyperbolic stochastic partial differential equations. Bates et al.  proved the existence of random attractors for stochastic reaction-diffusion equations on unbounded domains, and Wang and Tang [10, 11] described the properties of the random attractors.
Meanwhile the sine-Gordon equation is an important model in physics; Fan  considered the random attractor for the stochastic sine-Gordon equation. What are the other properties of stochastic sine-Gordon equation? As we know, Coayla-Teran  and Liu et al.  studied the mild solution of stochastic fractional partial differential equation with fractional and jump noises and considered the strong probability solution driven by white or Lévy noise for the stochastic nonlinear nonlocal parabolic equation and 2D stochastic N-S equation. In deterministic coupled case, the well-posedness of the solution for the nonlinear thermoelastic coupled sine-Gordon system has been studied by many authors, and the global attractor was treated in . Moreover, the more general thermoelastic system coupled model was considered. Gao and Muñoz Rivera  and Rivera  studied the well-posedness and energy decay rates. In deterministic case, nonlinear thermoelastic system coupled sine-Gordon equation is very weak coupling thermoelastic system; the more general model was investigated by Gao ; he considered the global attractor for the semilinear thermoelastic problem.
However, as far as we know, no one refers to the strong solution for stochastic nonlinear thermoelastic coupled sine-Gordon system by jump noise.
This paper is organized as follows. In the next section, we recall some fundamental results related to the solution of the stochastic equation and Lévy noise. In Section 3, we use the Galerkin method to prove the existence and uniqueness of solution to the problem (1). In Section 4, we give an asymptotic behavior of the solution of the problem (1).
In this paper, is a constant from line to line.
In this section, we recall some fundamental results related to some basic function spaces and the property of Lévy process; for more information, one can see [1, 2, 9, 12]. Let , with the domain , and are separable Hilbert spaces with the norm and , respectively; from Poincaré’s inequality, is equivalent to . Next, we recall some basic concepts related to Lévy process. The readers are referred to  for more details.
Let be a measurable space, and let be a -finite positive measure on it. If is a topological space, then by we will denote the Borel -field on , and is a Lebesgue measure on . Suppose that is a filtered probability space, where is a filtration and is a time homogeneous Poisson random measure with the intensity measure defined over the filtered probability space .
We will denote by the compensated Poisson random measure associated with , where the compensator is given by
We assume that is a Hilbert space. It is then known (see, for example, [1, 2]) that there exists a unique continuous linear operator which associates with each progressively measurable process satisfying
Moreover, is an -valued adapted and càdlàg process such that for any random step process satisfying the condition (3) with a representation where is a partition of , and for all , being an measurable random variable, one has In general case we write
The continuity (more precisely, isometry in Hilbert spaces) of the operator mentioned above means that
The class of all progressively measurable processes satisfying the condition (3) will be denoted by . If , the class of all progressively measurable processes satisfies the condition (3) just for this one , which will be denoted by .
The main technical tool in our paper is the It formula. Let us consider the Hilbert spaces and a -valued càdlàg process of the form where is a -valued process and is an -valued process; we have the following.
Theorem 1 (see [1, 2]). Suppose that and ; are progressively measurable processes. Suppose that is a -valued process given by (8) and there exists a -valued process such that in . Then is an -valued càdlàg -adapted process (up to distinguishable) and
3. Existence and Uniqueness of Solution
In this section, we use the Galerkin method to prove the local existence and uniqueness of solution; then making use of a priori estimates, we prove that there exists a convergence subsequence such that the solution is global.
As is well known, system (1) is equivalent to the following Itô system:
For simplicity, denote that , , and , . To obtain the existence of solution to (10), we suppose that the functions satisfy the following conditions: where and .
Definition 2. An -adapted stochastic process is said to be a strong probabilistic solution of stochastic nonlinear thermoelastic coupled system driven by Lévy noise (10) if it satisfies the following:(1) a.s. for any ;(2)the identities hold a.s. for all .
We now give our main result.
Theorem 3. Assume that is -adapted. The conditions and are satisfied. are the compensated Poisson random measure associated with , where the compensator is defined in Section 2. Then for any the stochastic nonlinear thermoelastic coupled system driven by Lévy noise (10) has a unique solution such that where is a positive constant.
Proof. Existence. We use the Galerkin approximation and some useful a priori estimates to prove the existence of solution. Set
where is the set of eigenfunctions of with domain ; it is an orthogonal set of and orthonormal one in , , , and . Denote by the orthogonal projector, where Span.
Hence, we can rewrite (10) as where, for each in .
Applying Itô formula to the process , we obtain
Denote that Since is an adapted and càdlàg process, the process is a martingale. Applying the Burkholder-Davis-Gundy inequality, condition , Hölder’s inequality, and Young’s inequality, we get Taking into account that the process has only positive jumps, we obtain From the Hlder inequality, we have
Putting (20)–(23) into (18), for , we have
Similarly, using the Itô formula to the process , we obtain Due to the Burkholder-Davis-Gundy inequality, condition , the Hölder inequality, and the Young inequality, Hence, we get
For every , (24) and (27) imply that where .
Then the Gronwall lemma implies that Substituting (29) into (28), we get
To complete the proof of the existence of solution we need to pass the limits in the Galerkin approximation. Owing to (29) and (30), there exists a subsequence of , not relabeled, such that From the conditions in , and (29)-(30), , we have Hence, there exist the functions such that Combining (31) and (34) and letting in (17), since the linear map is continuous from to (in fact an isometry), it is continuous with respect to the weak topologies. Therefore, in view of the weak convergence, we have for almost everywhere and .
Denote by the process which has a.s. sample paths being continuous in , is -adapted, and equals to almost everywhere ; then from (34) we obtain
Now, we consider the stopping time; for each , We claim that holds:
From (17) and (37), for any , we have a.s., for all .
For each , let Set , where is a positive constant to be defined later. By applying Itô’s formula to the processes , , and , respectively, we obtain Taking the mathematical expectations in (42) yields Let us analyze each term of (43). By the conditions and , we have Hence, using the conditions and again, we get
Substituting (44)–(46) into (43), we have where is the Sobolev embedding constant such that . Choosing , we get ; hence Replacing by in (48),
Since implies that , due to (31) and Sobolev embedding theorem, we have as ; hence then we have Similarly, we can prove that
Owing to (31)–(34) the sequence is bounded in , and by the Sobolev embedding theorem, Similarly, we can prove that
Therefore, the limits (51)–(54) imply that (39) holds true.
Next, due to the property of , we see that, for all , From this, since in , and (39), we have for almost everywhere .
Putting (56) into (37), we get
a.s., , .
From the property of , we obtain ; let and . For there exists such that for all .
Then, is a solution to (10).
Uniqueness. Set and as two solutions of (10); thus a.s., , .
From a similar argument as in the proof of existence, by the B-D-G inequality, conditions , and the Gronwall lemma, one can easily show that for ; thus for all .
We complete the proof of the theorem.
4. Asymptotic Behavior
In this section, we briefly discuss the long time behavior of the strong solutions of the stochastic nonlinear thermoelastic system coupled sine-Gordon equation driven by jump noise.
Following the idea in , we assume that there exists a constant such that