Pth Moment Exponential Stability of Nonlinear Hybrid Stochastic Heat Equations

Xuetao Yang,Zhangsong Yao,Quanxin Zhu

doi:10.1155/2014/481246

Abstract

We are concerned with the exponential stability problem of a class of nonlinear hybrid stochastic heat equations (known as stochastic heat equations with Markovian switching) in an infinite state space. The fixed point theory is utilized to discuss the existence, uniqueness, andpth moment exponential stability of the mild solution. Moreover, we also acquire the Lyapunov exponents by combining the fixed point theory and the Gronwall inequality. At last, two examples are provided to verify the effectiveness of our obtained results.

Highlights

There have been enormous theories for both definite and stochastic partial differential equations (SPDEs) since they can describe various phenomena in reality recently
We provide two examples of heat equations with Markovian switching as the applications of our main results and give some remarks compared to the previous articles
We have studied the stability problem of a class of nonlinear hybrid stochastic heat equations

Summary

Introduction

There have been enormous theories for both definite and stochastic partial differential equations (SPDEs) since they can describe various phenomena in reality recently. In [10], Bao et al discussed the pth moment exponential stability of linear hybrid stochastic heat equations by employing the explicit formulae and large deviation technique. We will use the fixed point principle to discuss the existence and uniqueness of the mild solution to (1) and prove the exponential stability result. In Theorem 5, we apply the fixed point theory to obtain the existence and uniqueness of the solution for a class of nonlinear hybrid stochastic heat equations. We apply succesfully the fixed point principle to study the pth moment exponential stability of a class of nonlinear hybrid stochastic heat equations, which obviously generalizes the linear case

Two Examples

Conclusion