一类非线性随机四阶抛物型方程的解的p阶矩指数稳定性

pth Moment Exponential Stability of a Class of Nonlinear Hybrid Stochastic Fourth-Order Parabolic Equations

本文讨论一类非线性随机四阶拋物型方程的解的P阶矩指数稳定性.{？u（t,x）/？t=Au（t,x）-A2u（t,x）＋α（r（t））▽k·f（t,u（t,x））,r（t））＋β（r（t））g（t,u（t,x））,r（t））B（t）u（t,x）=0 x∈？Θ,t＞0,u（0,x）=u0（x）,x∈Θ利用不动点原理,我们证明了方程的温和解的存在唯一性及P阶矩指数稳定性.

Stochastic partial differential equations have been paid more and more attention by mathematicians in the recent years because they are widely used in various fields. But there are few published research for the stochastic fourth-order parabolic equations. This situation motivates our present research. In this paper, we are mainly concerned with a class of nonlinear hybrid stochastic fourth-order parabolic equations.{？u（t,x）/？t=Au（t,x）-A2u（t,x）＋α（r（t））▽k·f（t,u（t,x））,r（t））＋β（r（t））g（t,u（t,x））,r（t））B（t）u（t,x）=0 x∈？Θ,t〉0,u（0,x）=u0（x）,x∈Θ. Under some suitable assumptions, we prove the existence, uniqueness, and pth moment exponential stability of the mild solution by using fixed point theorem. Our results extend and improve the conclusions in the previous literatures which are about the stability of the stochastic heat equations or the linear case of the stochastic fourth-order parabolic equations with Markov chains.