Asymptotic properties of some space-time fractional stochastic equations

Abstract

Consider non-linear time-fractional stochastic heat type equations of the following type, $$\partial^β_tu_t(x)=-ν(-Δ)^{α/2} u_t(x)+I^{1-β}_t[λσ(u)\stackrel{\cdot}{F}(t,x)]$$ in $(d+1)$ dimensions, where $ν>0, β\in (0,1)$, $α\in (0,2]$. The operator $\partial^β_t$ is the Caputo fractional derivative while $-(-Δ)^{α/2} $ is the generator of an isotropic stable process and $I^{1-β}_t$ is the fractional integral operator. The forcing noise denoted by $\stackrel{\cdot}{F}(t,x)$ is a Gaussian noise. And the multiplicative non-linearity $σ$ is assumed to be globally Lipschitz continuous. Under suitable conditions on the initial function, we study the asymptotic behaviour of the solution with respect to time and the parameter $λ$. In particular, our results are significant extensions of existing results. Along the way, we prove a number of interesting properties about the deterministic counterpart of the equation.