Dynamics of Non-Autonomous Stochastic Semi-Linear Degenerate Parabolic Equations with Nonlinear Noise

Abstract

In the present paper, we aim to study the long-time behavior of a stochastic semi-linear degenerate parabolic equation on a bounded or unbounded domain and driven by a nonlinear noise. Since the theory of pathwise random dynamical systems cannot be applied directly to the equation with nonlinear noise, we first establish the existence of weak pullback mean random attractors for the equation by applying the theory of mean-square random dynamical systems; then, we prove the existence of (pathwise) pullback random attractors for the Wong–Zakai approximate system of the equation. In addition, we establish the upper semicontinuity of pullback random attractors for the Wong–Zakai approximate system of the equation under consideration driven by a linear multiplicative noise.

Keywords

• stochastic degenerate parabolic equation
• nonlinear noise
• pullback random attractor
• Wong–Zakai approximation
• upper semicontinuity