The inhomogeneous fractional stochastic heat equation driven by fractional Brownian motion

We investigate the fractional Hardy-Hénon equation with fractional Brownian noise $${\mathrm{\partial }}_{t}u(t)+(-\mathrm{\Delta }{)}^{\theta \mathrm{/}2}u(t)=\mathrm{\mid }x{\mathrm{\mid }}^{-\gamma }\mathrm{\mid }u(t){\mathrm{\mid }}^{p-1}u(t)+\mu {\mathrm{\partial }}_t{B}^H(t),$$ 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 $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ 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.