The inhomogeneous fractional stochastic heat equation driven by fractional Brownian motion

Abstract

We investigate the fractional Hardy-Hénon equation with fractional Brownian noise $${\partial }_{t}u(t)+(-\Delta {)}^{\theta /2}u(t)=|x{|}^{-\gamma }|u(t){|}^{p-1}u(t)+\mu {\partial }_t{B}^H(t),$$ where $\theta >0$, $p>1$, $\gamma \ge 0$, $\mu \in \mathbb{R}$, and the random forcing $B^H$ is the fractional Brownian motion defined on some complete probability space $(\Omega ,\mathcal{F},\mathbb{P})$ with Hurst parameter $H\in (0,1)$. We establish the local existence and uniqueness of mild solutions under appropriate conditions on the parameters of the equation.

To appear in Arab Journal of Mathematical Sciences