Some properties of non-linear fractional stochastic heat equations on bounded domains

TL;DR

This paper investigates the properties of solutions to non-linear fractional stochastic heat equations on bounded domains, focusing on how solutions behave as the noise intensity parameter varies, extending previous results to more general operators and noise types.

Contribution

It extends existing analysis of stochastic heat equations to include non-local operators and colored spatial noise, providing new insights into solution behavior with respect to noise intensity.

Findings

Analysis of solution behavior as noise parameter varies

Extension to non-local operators and colored noise

Generalization of previous results in stochastic PDEs

Abstract

Consider the following stochastic partial differential equation, \begin{equation*} \partial_t u_t(x)= \mathcal{L}u_t(x)+ \xi\sigma (u_t(x)) \dot F(t,x), \end{equation*} where $\xi$ is a positive parameter and $\sigma$ is a globally Lipschitz continuous function. The stochastic forcing term $\dot F(t,x)$ is white in time but possibly colored in space. The operator $\mathcal{L}$ is a non-local operator. We study the behaviour of the solution with respect to the parameter $\xi$, extending the results in \cite{FoonNual} and \cite{Bin}