On some properties of a class of fractional stochastic heat equations

Abstract

We consider nonlinear parabolic stochastic equations of the form $\partial_t u=\sL u + λσ(u)\dot ξ$ on the ball $B(0,\,R)$, where $\dot ξ$ denotes some Gaussian noise and $σ$ is Lipschitz continuous. Here $\sL$ corresponds to an $α$-stable process killed upon exiting $B(0, R)$. We will consider two types of noise; space-time white noise and spatially correlated noise. Under a linear growth condition on $σ$, we study growth properties of the second moment of the solutions.