Space-time fractional stochastic partial differential equations with L\'evy Noise

Abstract

We consider non-linear time-fractional stochastic heat type equation $$\frac{\partial^\beta u}{\partial t^\beta}+\nu(-\Delta)^{\alpha/2} u=I^{1-\beta}_t \bigg[\int_{\mathbb{R}^d}\sigma(u(t,x),h) \stackrel{\cdot}{\tilde N }(t,x,h)\bigg]$$ and $$\frac{\partial^\beta u}{\partial t^\beta}+\nu(-\Delta)^{\alpha/2} u=I^{1-\beta}_t \bigg[\int_{\mathbb{R}^d}\sigma(u(t,x),h) \stackrel{\cdot}{N }(t,x,h)\bigg]$$ in $(d+1)$ dimensions, where $\alpha\in (0,2]$ and $d<\min\{2,\beta^{-1}\}\alpha$, $\nu>0$, $\partial^\beta_t$ is the Caputo fractional derivative, $-(-\Delta)^{\alpha/2} $ is the generator of an isotropic stable process, $I^{1-\beta}_t$ is the fractional integral operator, ${N}(t,x)$ are Poisson random measure with $\tilde{N}(t,x)$ being the compensated Poisson random measure. $\sigma:{\mathbb{R}}\to{\mathbb{R}}$ is a Lipschitz continuous function. We prove existence and uniqueness of mild solutions to this equation. Our results extend the results in the case of parabolic stochastic partial differential equations obtained in "M. Foondun and D. Khoshnevisan. Intermittence and nonlinear parabolic stochastic partial differential equations. \emph{ Electron. J. Probab.} {\bf14} (2009), 548--568" and " J. B. Walsh. An Introduction to Stochastic Partial Differential Equations, \'Ecoled'\'et\'e de Probabilit\'es de Saint-Flour, XIV|1984, Lecture Notes in Math., vol. 1180, Springer, Berlin, (1986), 265--439". Under the linear growth of $\sigma$, we show that the solution of the time fractional stochastic partial differential equation follows an exponential growth with respect to the time. We also show the nonexistence of the random field solution of both stochastic partial differential equations when $\sigma$ grows faster than linear.