Higher-Order Continuity of Pullback Random Attractors for Random Quasilinear Equations with Nonlinear Colored Noise

For a nonautonomous random dynamical system, we introduce a concept of a pullback random bi-spatial attractor (PRBA). We prove an existence theorem of a PRBA, which includes its measurability, compactness and attraction in the regular space. We then establish the residual dense continuity of a family of PRBAs from a parameter space into the space of all compact subsets of the regular space equipped by Hausdorff metric. The abstract results are illustrated in the nonautonomous random quasilinear equation driven by nonlinear colored noise, where the size of noise belongs to \((0,\infty ]\) $(0,\infty ]$ and the infinite size corresponds to the deterministic equation. The application results are the existence and residual dense continuity of PRBAs on \((0,\infty ]\) $(0,\infty ]$ in both square and p-order Lebesgue spaces, where \(p>2\) $p>2$ . The lower semi-continuity of attractors in the regular space seems to be a new subject even for an autonomous deterministic system.