Intermittence and time fractional stochastic partial differential equations

We consider time fractional stochastic heat type equation $${\mathrm{\partial }}_t^{\beta }u(t,x)=-\nu (-\mathrm{\Delta }{)}^{\alpha \mathrm{/}2}{u}_t(x)+I_t^{1-\beta }[\sigma (u)(t,x)]$$ in $(d+1)$ dimensions, where $\nu >0$, $\beta \in (0,1)$, $\alpha \in (0,2]$, $d<\min \{2,{\beta }^{-1}\}\text{a}$, ${\mathrm{\partial }}_t^{\beta }$ is the Caputo fractional derivative, $-(-\mathrm{\Delta }{)}^{\alpha \mathrm{/}2}$ is the generator of an isotropic stable process, $(t,x)$ is space-time white noise, and $\sigma :\text{RR}R\to \text{RR}R$ is Lipschitz continuous. The time fractional stochastic heat type equations might be used to model phenomenon with random effects with thermal memory. We prove: (i) absolute moments of the solutions of this equation grows exponentially; and (ii) the distances to the origin of the farthest high peaks of those moments grow exactly linearly with time. These results extend the results of Foondun and Khoshnevisan \cite{foondun-khoshnevisan-09} %(Mohammud Foondun and Davar Khoshnevisan, Intermittence and nonlinear parabolic %stochastic partial differential equations, Electron. J. Probab. 14 (2009), no. 21, 548–568) and Conus and Khoshnevisan \cite{conus-khoshnevisan} % (On the existence and position of the farthest peaks of a family of stochastic %heat and wave equations, Probab. Theory Related Fields 152 (2012), no. 3-4, 681–701) on the parabolic stochastic heat equations.