Nonlinear random perturbations of Reaction-Diffusion Equations

TL;DR

This paper studies the well-posedness and small-noise behavior of nonlinear stochastic reaction-diffusion equations with non-local diffusion coefficients depending on the solution's conditional expectation, addressing analytical challenges from non-locality and low regularity.

Contribution

It extends the analysis of nonlinear SPDEs by establishing well-posedness and asymptotics under minimal regularity assumptions, including non-local and nonlinear diffusion.

Findings

Proved well-posedness of the class of nonlinear SPDEs.

Derived small-noise asymptotic behavior of solutions.

Addressed analytical challenges from non-local diffusion and low regularity.

Abstract

This paper investigates the well-posedness and small-noise asymptotics of a class of stochastic partial differential equations defined on a bounded domain of $\mathbb{R}^d$, where the diffusion coefficient depends nonlinearly and non-locally on the solution through a conditional expectation. The reaction term is assumed to be merely continuous and to satisfy a quasi-dissipativity condition, without requiring any growth bounds or local Lipschitz continuity. This setting introduces significant analytical challenges due to the temporal non-locality and the lack of regularity assumptions. Our results represent a substantial advance in the study of nonlinear stochastic perturbations of SPDEs, extending the framework developed in a previous paper.