Limiting Behavior of Invariant Measures for Stochastic Quasilinear Parabolic Equations with Nonlinear Noise on Thin Domains

In this paper, the limiting behavior of invariant measures is mainly investigated for a class of stochastic quasilinear parabolic equations with nonlinear noise on thin domains. The existence and uniqueness of invariant measure on \((n+1)\)-dimensional thin domains are presented. The difficulty on estimates of the solutions for such problems in Sobolev space in the sense of thin domains is overcome by a novel proof techniques. Hence, the research results reveal that any limit of invariant measures of original equations on thin domains must be an invariant measure of the limiting equations when the \((n+1)\)-dimensional thin domains degenerates onto the n-dimensional space.