Blow-up results for space-time fractional stochastic partial differential equations

Abstract

Consider non-linear time-fractional stochastic reaction-diffusion equations of the following type, $${\partial }_t^{\beta }{u}_t(x)=-\nu (-\Delta {)}^{\alpha /2}{u}_t(x)+I_t^{1-\beta }[b(u)+\sigma (u)\stackrel{\cdot }{F}(t,x)]$$ in $(d+1)$ dimensions, where $\nu >0,\beta \in (0,1)$, $\alpha \in (0,2]$. The operator ${\partial }_t^{\beta }$ is the Caputo fractional derivative while $-(-\Delta {)}^{\alpha /2}$ is the generator of an isotropic $\alpha$-stable Lévy process and $I_t^{1-\beta }$ is the Riesz fractional integral operator. The forcing noise denoted by $\stackrel{\cdot }{F}(t,x)$ is a Gaussian noise. These equations might be used as a model for materials with random thermal memory. We derive non-existence (blow-up) of global random field solutions under some additional conditions, most notably on $b$, $\sigma$ and the initial condition. Our results complement those of P. Chow in \cite{chow2}, \cite{chow1}, and Foondun et al. in \cite{Foondun-liu-nane}, \cite{foondun-parshad} among others.

38 pages, Submitted for publication. arXiv admin note: text overlap with arXiv:1611.07282