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

Abstract

We consider time fractional stochastic heat type equation $${\partial }_t^{\beta }{u}_t(x)=-\nu (-\Delta {)}^{\alpha /2}{u}_t(x)+I_t^{1-\beta }[\sigma (u)\stackrel{\cdot }{W}(t,x)]$$ in $(d+1)$ dimensions, where $\nu >0$, $\beta \in (0,1)$, $\alpha \in (0,2]$, $d<\min\{2,β^{-1}\}\a$, ${\partial }_t^{\beta }$ is the Caputo fractional derivative, $-(-\Delta {)}^{\alpha /2}$ 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 $\alpha =2$ and $d=1.$ In this paper we extend this result to the case $\alpha =2$ and $d\in \{1,2,3\}.$

arXiv comment17 pages, submitted for publication. arXiv admin note: substantial text overlap with arXiv:1409.7468

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