Intermittency fronts for space-time fractional stochastic partial differential equations in $(d+1)$ dimensions

We consider time fractional stochastic heat type equation $\partial^β_tu_t(x)=-ν(-Δ)^{α/2} u_t(x)+I^{1-β}_t[σ(u)\stackrel{\cdot}{W}(t,x)]$ in $(d+1)$ dimensions, where $ν>0$, $β\in (0,1)$, $α\in (0,2]$, $d<\min\{2,β^{-1}\}\a$, $\partial^β_t$ is the Caputo fractional derivative, $-(-Δ)^{α/2}$ is the generator of an isotropic stable process, $\stackrel{\cdot}{W}(t,x)$ is space-time white noise, and $σ:\R \to\RR{R}$ is Lipschitz continuous. Mijena and Nane proved in \cite{JebesaAndNane1} that : (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. The last result was proved under the assumptions $α=2$ and $d=1.$ In this paper we extend this result to the case $α=2$ and $d\in\{1,2,3\}.$

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