Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations

A concept of a bi-spatial random attractor for a random dynamical system is introduced. A unified result about existence and upper semi-continuity for a family of bi-spatial random attractors is obtained if a family of random systems is convergent, uniformly absorbing in an initial space and uniformly omega-compact in both initial and terminate spaces. The upper semi-continuity result improves all existing results even for single-spatial attractors. As an application of the abstract result, it is shown that every semilinear Laplacian equation on the entire space perturbed by a multiplicative and stochastic noise possesses an (L2,Lq)(L2,Lq)-random attractor with q>2q>2. Moreover, it is proved that the family of obtained attractors is upper semi-continuous at any density of noises and the family of attractors for the corresponding compact systems is both upper and lower semi-continuous at infinity under the topology of both spaces.