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

TL;DR

This paper extends the analysis of space-time fractional stochastic heat equations, demonstrating linear growth of high peaks' distance in higher dimensions for the case =2, building on previous work limited to one dimension.

Contribution

The paper generalizes previous results on intermittency fronts to include dimensions 1, 2, and 3 for the case =2, expanding understanding of solution behaviors in higher-dimensional stochastic PDEs.

Findings

High peaks of solutions grow linearly with time in higher dimensions.

Exponential growth of solution moments is confirmed in extended dimensions.

Results extend previous one-dimensional analysis to three dimensions.

Abstract

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 fractional derivative, $-(-\Delta)^{\alpha/2} $ is the generator of an isotropic stable process, $\stackrel{\cdot}{W}(t,x)$ is space-time white noise, and $\sigma:\R \to\RR{R}$ 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 $d\in\{1,2,3\}.$