Some properties of non-linear fractional stochastic heat equations on bounded domains

We consider the following fractional stochastic partial differential equation on a bounded, open subset B of Rd for d ≥ 1 ∂tut(x)=Lut(x)+ξσ(ut(x))F˙(t,x),where ξ is a positive parameter and σ is a globally Lipschitz continuous function. The stochastic forcing term F˙(t,x) is white in time but possibly colored in space. The operator L is fractional Laplacian which is the infinitesimal generator of a symmetric α-stable Lévy process in Rd. We study the behaviour of the solution with respect to the parameter ξ.We show that under zero exterior boundary conditions, in the long run, the pth-moment of the solution grows exponentially fast for large values of ξ. However when ξ is very small we observe eventually an exponential decay of the pth-moment of this same solution.