Pth moment exponential stability of hybrid stochastic fourth-order parabolic equations

Ling Wei,Quanxin Zhu,Mianjun Duan,Zhangsong Yao

doi:10.1186/s13662-016-0781-y

Abstract

We are concerned with a class of hybrid stochastic fourth-order parabolic equations with Markov switching in an infinite state space. By employing the fixed point theory we study the existence, uniqueness and pth moment exponential stability of the mild solution. Finally, we provide two examples to verify the effectiveness of our results.

Highlights

In recent years, the hybrid stochastic differential equations (SDEs) and hybrid stochastic partial differential equations (SPDEs), for example, hybrid stochastic heat equations, have been paid much attention owing to their wide applications in natural science, engineering, biology, finance, and other areas
The Lyapunov method provides a powerful implement in studying the stability of SDEs, and there are many methods to study the stability of SPDEs including hybrid stochastic heat equations, such as the Lyapunov function method [ ], successive approximation approach [ ], large deviation technique [, ], fixed pointed theory [ – ], and so on
It should be mentioned that the fixed point theory, which is introduced by Burton, is a very important method to discuss the stability of both deterministic and stochastic differential equations

Summary

Introduction

The hybrid stochastic differential equations (SDEs) and hybrid stochastic partial differential equations (SPDEs), for example, hybrid stochastic heat equations, have been paid much attention owing to their wide applications in natural science, engineering, biology, finance, and other areas. As a model to describe these phenomena, the Cahn-Hilliard equation has intrigued many mathematicians’ interests, and many good results [ – ] were obtained, such as the global existence, the asymptotic behavior, the stability of the solution of the Cahn-Hilliard equation, and so on. Inspired by the method of fixed point theory, which is widely used in the discussion of hybrid stochastic heat equations, in this paper, we are concerned with the stability problem for a class of hybrid stochastic fourth-order parabolic equations. In Section , based on the basic solution of the definite homogeneous fourth-order parabolic equation, by applying the fixed point theory we prove the existence, uniqueness, and pth moment exponential stability of hybrid stochastic fourth-order parabolic equations. In Section , we provide two examples to verify the effectiveness of the obtained results with some general remarks

Preliminaries

Conclusions