Random attractors for degenerate stochastic partial differential equations

TL;DR

This paper establishes the existence of random attractors for a broad class of degenerate stochastic partial differential equations with rough noise, using a variational approach and stationary solutions, with applications to porous media and reaction diffusion equations.

Contribution

It introduces a novel method for proving the existence of random attractors under minimal assumptions, accommodating rougher noise than previous results.

Findings

Existence of random attractors for degenerate SPDEs with additive and multiplicative noise.

Collapse of deterministic attractors to a single random point under sufficient noise.

Applicability to porous media, p-Laplace, and reaction diffusion equations.

Abstract

We prove the existence of random attractors for a large class of degenerate stochastic partial differential equations (SPDE) perturbed by joint additive Wiener noise and real, linear multiplicative Brownian noise, assuming only the standard assumptions of the variational approach to SPDE with compact embeddings in the associated Gelfand triple. This allows spatially much rougher noise than in known results. The approach is based on a construction of strictly stationary solutions to related strongly monotone SPDE. Applications include stochastic generalized porous media equations, stochastic generalized degenerate p-Laplace equations and stochastic reaction diffusion equations. For perturbed, degenerate p-Laplace equations we prove that the deterministic, infinite dimensional attractor collapses to a single random point if enough noise is added.