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 k p − 2 k + 2 order. This higher-order integrability shows continuity of the solution operator from the square Lebesgue space to ( k p − 2 k + 2 ) -order Lebesgue spaces. In particular, the ( 2 p − 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.