Large time behaviour for the non-linear fractional stochastic heat equation

Mohammud Foondun, Ngartelbaye Guerngar, Erkan Nane

Published 2016-05-04Version 1

We consider the following fractional stochastic partial differential equation on $D$ an open bounded subset of $\mathbb{R}^d$ for $d\geq 1$ \begin{equation*} \partial_t u_t(x)= -\frac{1}{2}(-\Delta)^{\frac{\alpha}{2}} u_t(x)+ \xi\sigma (u_t(x)) \dot W(t,x) \ \ \text{for} \ \alpha \in (0,2] \end{equation*} where the fractional Laplacian is the infinitesimal generator of a symmetric $\alpha$-stable process in $\mathbb{R}^d$ and can be written in the form \begin{align*} -(-\Delta)^{\frac{\alpha}{2}} u(x)= c \lim\limits_{\varepsilon \searrow 0}\int_{\{y\in\mathbb{R}^d: |y-x|>\varepsilon\}} (u(y)-u(x))\frac{dy}{|y-x|^{d+\alpha}} \end{align*} for some constant $c= c(\alpha, d)$. $\xi$ is a parameter in $\mathbb{R}$, $\sigma$ is a Lipschitz continuous function and $\dot W(t,x)$ is a Gaussian noise white in time and white or coloured in space. We show that under Dirichlet conditions, in the long run, the $p^{\text{th}}$-moment of the solution grows exponentially fast for large values of $\xi$. However when $\xi$ is very small we observe eventually an exponential decay of the $p^{\text{th}}$-moment of this same solution. It is important to recall here that the case $\alpha=2$ is provided in \cite{f}. Therefore we will consider $\alpha\in(0,2)$.