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 $u\left(t,x\right)$ can not decay exponentially and for $\beta \in \left(0,\frac{1}{2}\right]$, ${sup}_{x\in D}\mathbb{𝔼}|u\left(t,x\right){|}^2$ grows exponentially fast for large $t$. When $\beta \in \left(\frac{1}{2},1\right)$, there is some phase transition of the second moment growth, depending on the noise level $\lambda$.