Random attractors for singular stochastic partial differential equations

TL;DR

This paper proves the existence of random attractors for singular SPDEs with additive noise and explores their properties, including finite-time extinction under multiplicative noise, with applications to various stochastic PDEs.

Contribution

It establishes the existence of random attractors for a broad class of singular SPDEs under minimal assumptions and analyzes their structure and extinction properties.

Findings

Existence of random attractors for singular SPDEs with additive noise.

Single random point attractor for ergodic, monotone systems.

Finite-time extinction for equations with linear multiplicative noise.

Abstract

The existence of random attractors for singular stochastic partial differential equations (SPDE) perturbed by general additive noise is proven. The drift is assumed only to satisfy the standard assumptions of the variational approach to SPDE with compact embeddings in the Gelfand triple and singular coercivity. For ergodic, monotone, contractive random dynamical systems it is proven that the attractor consists of a single random point. In case of real, linear multiplicative noise finite time extinction is obtained. Applications include stochastic generalized fast diffusion equations and stochastic generalized singular p-Laplace equations perturbed by Levy noise with jump measure having finite first and second moments.