Continuous Wong–Zakai Approximations of Random Attractors for Quasi-linear Equations with Nonlinear Noise

Abstract

We consider a family of random quasi-linear equations driven by nonlinear Wong–Zakai noise and parameterized by the non-zero size $$\lambda $$ of noise. After proving the existence of a random attractor $$A_\lambda (\omega )$$ in the square Lebesgue space, we then show that there is a residual dense subset of the space of nonzero real numbers such that, under the Hausdorff metric, the map $$\lambda \rightarrow A_\lambda (\theta _s\omega )$$ is continuous at all points of the residual dense set, where $$\theta _s$$ is a group of self-transformations on the probability space. We also prove that as $$\lambda \rightarrow \pm \infty $$ the random attractor converges upper-semicontinuously to the global attractor of the deterministic quasi-linear equation. The upper semi-continuity result is new for nonlinear noise, while, the lower semi-continuity result is new even for linear noise. The theory of Baire category is the main tool used to prove the residual continuity.