Some properties of non-linear fractional stochastic heat equations on bounded domains. (September 2017) - Foondun, Mohammud
Abstract: We consider the following fractional stochastic partial differential equation on a bounded, open subset B of R d for d ≥ 1 ∂ t u t ( x ) = L u t ( x ) + ξ σ ( u t ( 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 R d . 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 p th-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 p th-moment of this same solution.