A note on intermittency for the fractional heat equation

The goal of the present note is to study intermittency properties for the solution to the fractional heat equation ∂u∂t(t,x)=−(−Δ)β/2u(t,x)+u(t,x)Ẇ(t,x),t>0,x∈Rd with initial condition bounded above and below, where β∈(0,2]β∈(0,2] and the noise WW behaves in time like a fractional Brownian motion of index H>1/2H>1/2, and has a spatial covariance given by the Riesz kernel of index α∈(0,d)α∈(0,d). As a by-product, we obtain that the necessary and sufficient condition for the existence of the solution is α<βα<β.