Nonlinear fractional stochastic heat equation driven by Gaussian noise rough in space

Abstract

In this paper we consider a class of nonlinear fractional stochastic heat equation∂∂tu(t,x)=Dδαu(t,x)+σ(u(t,x))W˙(t,x),(t,x)∈[0,T]×R, in (1+1)-dimension with T>0, where Dδα is a nonlocal fractional differential operator with α∈(1,2) and |δ|≤2−α. The diffusion coefficient σ(⋅) is a measurable function. W˙ is a Gaussian noise which is white in time and behaves as a fractional Brownian motion with Hurst index H satisfying 3−α4<H<12 in the space variable. Under some mild assumptions, we prove the existence and uniqueness of the mild solution in some function spaces. Along the way, we study the moment bounds for the solution and show that the solution is weakly full intermittent.