Space-time fractional stochastic partial differential equations driven by Lévy white noise

Abstract

This paper is concerned with the following space-time fractional stochastic nonlinear partial differential equation \begin{equation*} \left(\partial_t^β+\fracν{2}\left(-Δ\right)^{α/ 2}\right) u=I_{t}^γ\Big[ f(t,x,u)-\sum_{i=1}^{d} \frac{\partial}{\partial x_i} q_i(t,x,u)+ σ(t,x,u) F_{t,x}\Big] \end{equation*} for a random field $u(t,x):[0,\infty )\times {\mathbb{R}}^d\mapsto \mathbb{R}$, where $\hat{I}\pm >0,{\hat{I}}^2\in (0,2),{\hat{I}}^3\ge 0,\hat{I}½>0,F_{t,x}$ is a Lévy space-time white noise, $I_t^3$ stands for the Riemann-Liouville integral in time, and $f,q_i,\ddot{I}ƒ:[0,\infty )\times {\mathbb{R}}^d\times \mathbb{R}\mapsto \mathbb{R}$ are measurable functions. Under suitable polynomial growth conditions, we establish the existence and uniqueness of $L^2({\mathbb{R}}^d)$-valued local solutions when the Lévy white noise $F_{t,x}$ contains Gaussian noise component. Furthermore, for $p\in [1,2]$, we derive the existence and uniqueness of $L^p({\mathbb{R}}^d)$-valued local solutions for the equation driven by pure jump Lévy white noise. Finally, we obtain certain stronger conditions for the existence and uniqueness of global solutions.