Abstract and Applied Analysis

Volume 2014 (2014), Article ID 403528, 12 pages

http://dx.doi.org/10.1155/2014/403528

Research Article

Stochastic Nonlinear Thermoelastic System Coupled Sine-Gordon Equation Driven by Jump Noise

Shuilin Cheng,¹ Yantao Guo,1,2 and Yanbin Tang¹

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.

Abstract.

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.

1. Introduction

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 [5]).

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. [8] studied the existence of invariant manifolds for coupled parabolic and hyperbolic stochastic partial differential equations. Bates et al. [9] 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 [12] 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 [13] and Liu et al. [3] 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 [14]. Moreover, the more general thermoelastic system coupled model was considered. Gao and Muñoz Rivera [15] and Rivera [16] 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 [17]; 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.

2. Preliminaries

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 [1] 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 [6], we assume that there exists a constant such that

where is defined in Section 3.

Theorem 4. Suppose that the conditions for Theorem 3 hold true. If , then the solution of the problem (10) satisfies

where are the positive constants.

Proof. Set , using Itô’s formula to the processes , , and , respectively; we obtain

Taking the mathematical expectations in (63) yields

From Hölder’s inequality, Young’s inequality, and condition , we have

Therefore, from (64), we get

Due to the embedding theorem and (61),

Hence, by Theorem 3, we have

Gronwall’s inequality leads to

With the choice of , the assumption of Theorem 3 holds true; we obtain that

This completes the proof of the theorem.

Remark 5. Since denotes the displacement at point on an orbit , a.s., means the velocity; the result in the Theorem 4 exhibits that the velocity is exponentially decay in time in the sense of mean square; in the view of physics, one can obtain that the displacement will tend to a constant in the large time in the sense of mean square.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are grateful to the anonymous referees for helpful comments and suggestions that greatly improved the presentation of this paper. This work is supported by NSF of China (11272277 and 11226188) and FRF for the Central Universities of China (2013ZZGH027).

References

D. Applebaum, Lévy Processes and Stochastic Calculus, vol. 116, Cambridge University Press, Cambridge, UK, 2nd edition, 2009.

Z. Brzeźniak, E. Hausenblas, and J. Zhu, “2D stochastic Navier-Stokes equations driven by jump noise,” Nonlinear Analysis, vol. 79, pp. 122–139, 2013.

J. Liu, L. Yan, and Y. Cang, “On a jump-type stochastic fractional partial differential equation with fractional noises,” Nonlinear Analysis, vol. 75, no. 16, pp. 6060–6070, 2012.

P. A. Razafimandimby and M. Sango, “Strong solution for a stochastic model of two-dimensional second grade fluids: existence, uniqueness and asymptotic behavior,” Nonlinear Analysis, vol. 75, no. 11, pp. 4251–4270, 2012.

R. Chiba and Y. Sugano, “Stochastic thermoelastic problem of a functionally graded plate under random temperature load,” Archive of Applied Mechanics, vol. 77, pp. 215–227, 2007.

P.-L. Chow, Stochastic Partial Differential Equations, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2007.

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, UK, 1996.

T. Caraballo, I. Chueshov, and J. A. Langa, “Existence of invariant manifolds for coupled parabolic and hyperbolic stochastic partial differential equations,” Nonlinearity, vol. 18, no. 2, pp. 747–767, 2005.

P. W. Bates, K. Lu, and B. Wang, “Random attractors for stochastic reaction-diffusion equations on unbounded domains,” Journal of Differential Equations, vol. 246, no. 2, pp. 845–869, 2009.

G. Wang and Y. B. Tang, “ random attractors for stochastic reaction-diffusion equation on unbounded domains,” Abstract and Applied Analysis, vol. 2013, Article ID 279509, 23 pages, 2013.

G. Wang and Y. B. Tang, “Fractal dimension of a random invariant set and applications,” Journal of Applied Mathematics, vol. 2013, Article ID 415764, 5 pages, 2013.

X. Fan, “Random attractor for a damped sine-Gordon equation with white noise,” Pacific Journal of Mathematics, vol. 216, no. 1, pp. 63–76, 2004.

E. A. Coayla-Teran, “Strong solutions for stochastic nonlinear nonlocal parabolic equations,” Stochastics and Dynamics, vol. 10, no. 4, pp. 497–508, 2010.

J. W. Zhang, Y. H. Ren, R. H. Wu, and T. Feng, “The global attractor of nonlinear thermoelastic coupled sine-Gordon system,” Acta Physica Sinica, vol. 61, no. 11, Article ID 110404, 2012 (Chinese).

H. Gao and J. E. Muñoz Rivera, “Global existence and decay for the semilinear thermoelastic contact problem,” Journal of Differential Equations, vol. 186, no. 1, pp. 52–68, 2002.

J. E. M. Rivera, “Energy decay rates in linear thermoelasticity,” Funkcialaj Ekvacioj, vol. 35, no. 1, pp. 19–30, 1992.

H. Gao, “Global attractor for the semilinear thermoelastic problem,” Mathematical Methods in the Applied Sciences, vol. 26, no. 15, pp. 1255–1271, 2003.