Some properties of space-time fractional stochastic partial differential equations with Lévy noise

Abstract

We consider a class of space-time fractional stochastic partial differential equation on a bounded domain with Lévy noise. We prove that the second moment of the solution u(t,x) can not decay exponentially and for β∈(0,12], supx∈D𝔼|u(t,x)|2 grows exponentially fast for large t. When β∈(12,1), there is some phase transition of the second moment growth, depending on the noise level λ.