Limiting dynamics for stochastic nonclassical diffusion equations

Abstract

In this paper, we are concerned with the dynamical behavior of the stochastic nonclassical parabolic equation, more precisely, it is shown that the inviscid limits of the stochastic nonclassical diffusion equations reduces to the stochastic heat equations. We deal with initial values in $H^{1}_{0}(I)$ and $H^{2}(I)\cap H^{1}_{0}(I)$. When the initial value in $H^{1}_{0}(I),$ we establish the inviscid limits of the weak martingale solution; when the initial value in $H^{2}(I)\cap H^{1}_{0}(I),$ we establish the inviscid limits of the weak solution, the convergence in probability in $L^{2}(0,T;H^{1}(I))$ is proved.The results are valid for cubic nonlinearity. The key points in the proof of our convergence results are establishing some uniform estimates and the regularity theory for the solutions of the stochastic nonclassical diffusion equations which are independent of the parameter. Based on the uniform estimates, the tightness of distributions of the solutions can be obtained.