Approximate inertial manifolds of exponential order for semilinear parabolic equations subjected to additive white noise

Abstract

A class of semilinear nonautonomous parabolic equations subjected to additive white noise is considered. The existence of a family of randomN-dimensional approximate inertial manifolds (AIMs) whose neighborhoods of thickness of order exp(-δλ N+1 β ) attract exponentially in the mean all the trajectories is proved forN large enough. Hereλ N+1 is the (N+1)th eigenvalue of the corresponding linear problem, andδ andΒ are positive constants. We also construct a sequence of AIMs which converges to the exact inertial manifold, when a spectral gap condition is satisfied. These results remain true for deterministic autonomous and nonautonomous cases.