Intermittency fronts for space-time fractional stochastic partial differential equations in [formula omitted] dimensions

Abstract

We consider time fractional stochastic heat type equation ∂tβut(x)=−ν(−Δ)α/2ut(x)+It1−β[σ(u)W⋅(t,x)] in (d+1) dimensions, where ν>0, β∈(0,1), α∈(0,2], d<min{2,β−1}α, ∂tβ is the Caputo fractional derivative, −(−Δ)α/2 is the generator of an isotropic stable process, W⋅(t,x) is space-time white noise, and σ:R→R is Lipschitz continuous. Mijena and Nane proved in (Mijena and Nane, 2016) that: (i) absolute moments of the solutions of this equation grow 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∈{1,2,3}.