Long time behavior of stochastic parabolic problems with white noise in materials with thermal memory

Abstract

The existence and limiting behavior of the solutions of stochastic parabolic problems with thermal memory are investigate in the cases that the nonlinear term satisfies subcritical and critical growth conditions. The existence, uniqueness and continuity of solutions is proved by a semigroup method and the Lax-Milgram theorem, then the dynamics of solutions is analyzed by a priori estimates. In particular, the existence of pullback random attractors for the random dynamical system associated to the problem is established and the upper semi-continuity of the pullback random attractors is verified.