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We consider time fractional stochastic heat type equation$$\partial^\beta_tu(t,x)=-\nu(-\Delta)^{\alpha/2}u_t(x)+I^{1-\beta}_t[\sigma(u)\stackrel{\cdot}{W}(t,x)]$$ in $(d+1)$dimensions, where $\nu>0$, $\beta\in (0,1)$, $\alpha\in (0,2]$,$d<\min\{2,\beta^{-1}\}\a$, $\partial^\beta_t$ is the Caputo fractionalderivative, $-(-\Delta)^{\alpha/2} $ is the generator of an isotropic stableprocess, $\stackrel{\cdot}{W}(t,x)$ is space-time white noise, and$\sigma:\RR{R}\to\RR{R}$ is Lipschitz continuous.  The time fractional stochastic heat type equations might be used to modelphenomenon with random effects with thermal memory. We prove: (i) absolutemoments of the solutions of this equation grows exponentially; and (ii) thedistances to the origin of the farthest high peaks of those moments growexactly linearly with time. These results extend the results of Foondun andKhoshnevisan \cite{foondun-khoshnevisan-09}  %(Mohammud Foondun and Davar Khoshnevisan, Intermittence and nonlinearparabolic %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 ofstochastic %heat and wave equations, Probab. Theory Related Fields 152 (2012),no. 3-4, 681--701) on the parabolic stochastic heat equations.

Intermittence and time fractional stochastic partial differential equations