Existence, Stability and Bifurcation of Random Complete and Periodic Solutions of Stochastic Parabolic Equations

TL;DR

This paper investigates the existence, stability, and bifurcation phenomena of random solutions in stochastic parabolic equations with multiplicative noise, including periodic solutions and bifurcation analysis.

Contribution

It provides new results on the existence, uniqueness, and stability of random attractors and solutions, including bifurcation phenomena in stochastic parabolic equations.

Findings

Existence and uniqueness of tempered random attractors.

Characterization of attractors by random complete solutions.

Bifurcation and multiplicity of solutions in stochastic Chafee-Infante equation.

Abstract

In this paper, we study the existence, stability and bifurcation of random complete and periodic solutions for stochastic parabolic equations with multiplicative noise. We first prove the existence and uniqueness of tempered random attractors for the stochastic equations and characterize the structures of the attractors by random complete solutions. We then examine the existence and stability of random complete quasi-solutions and establish the relations of these solutions and the structures of tempered attractors. When the stochastic equations are incorporated with periodic forcing, we obtain the existence and stability of random periodic solutions. For the stochastic Chafee-Infante equation, we further establish the multiplicity and stochastic bifurcation of complete and periodic solutions.