Box-counting dimensions and upper semicontinuities of bi-spatial attractors for stochastic degenerate parabolic equations on an unbounded domain

The paper contributes box-counting dimensions and upper semicontinuities of random bi-spatial attractors for stochastic degenerate parabolic equations on the whole Euclid space. Under some weak assumptions for the force and the nonlinearity, we first prove the existence of a unique (L2,D01,2∩Lq)-random attractor for any q∈[2,(p−2)I+2], where p−1 is the order of the nonlinearity and I is a given integer such that the force is (I+1)-times integrable. By using truncation and splitting techniques, and also induction methods, we then prove that a priori estimate is uniform with respect to the density of noise, which leads to the upper semicontinuity result of the obtained attractors as the density tends to a constant (including zero) under the topology of the terminative space. Furthermore, we give a new framework to discuss the bound of Lq-box-counting dimensions of random attractors for SPDE on an unbounded domain.