Limiting dynamics for stochastic reaction–diffusion equations on the Sobolev space with thin domains

Abstract

This paper is devoted to bi-spatial random attractors of the stochastic reaction–diffusion equation when the terminate space is the Sobolev space on a thin domain, where the nonlinearity can be decomposed into two functions with (p,q)-growth exponents. By means of a computation method of induction, it is shown that the difference of solutions near the initial time is integrable of kp−2k+2 order. This higher-order integrability shows continuity of the solution operator from the square Lebesgue space to (kp−2k+2)-order Lebesgue spaces. In particular, the (2p−2)-order integrability shows continuity of the solution operator from the square Lebesgue space to the Sobolev space, which further shows existence of a random attractor in the Sobolev space when the initial space is the square Lebesgue space. Moreover, the higher-order integrability can be uniform with respect to all thin domains, which provides uniformly asymptotic compactness of the random dynamical systems. As a conclusion, the upper semi-continuity of attractors under the Sobolev norm is established when the narrow domain degenerates onto a lower dimensional domain.