Local upper semicontinuity of bispatial attractors for nonautonomous stochastic parabolic equations with singular perturbation

We study the local upper semicontinuity of pullback bispatial attractors for singularly perturbed nonautonomous stochastic reaction-diffusion equations on an unbounded domain. The local property means that the semicontinuity of bispatial attractors is local uniform with regard to the time as the singular perturbation parameter tends to 0. To do this, by introducing some new and local-uniform concepts of nonautonomous cocycles, we first establish the regularity and local upper semicontinuity of pullback attractors simultaneously. In applications, the presence of the singular perturbation term epsilon Delta 2u brings some difficulties even to prove the existence of pullback attractors in H1. Therefore, we prove an (L2,L2)-pullback attractor without any restriction on the order of the nonlinearity. With the critical exponent of the nonlinearity, we apply positive-negative truncations and splitting methods to attest an (L2,H1)-pullback attractor. We use weak assumptions for both the nonlinearity and force here. Finally, from the viewpoint of left and right limits, we prove the local upper semicontinuity of the family of obtained attractors as the perturbation parameter approaches to 0 in L2 & x22c2;H1, by using the continuity of solutions on the time. This result essentially improves and completes previous works on upper semicontinuity of attractors only as the change of single parameter.