Moment bounds of a class of stochastic heat equations driven by space–time colored noise in bounded domains

Abstract

We consider the fractional stochastic heat type equation ∂∂tut(x)=−(−Δ)α∕2ut(x)+ξσ(ut(x))Ḟ(t,x),x∈D,t>0,with nonnegative bounded initial condition, where α∈(0,2], ξ>0 is the noise level, σ:R→R is a globally Lipschitz function satisfying some growth conditions and the noise term Ḟ behaves in space like the Riez kernel and is possibly correlated in time and D is the unit open ball centered at the origin in Rd. When the noise term is not correlated in time, we establish a change in the growth of the solution of these equations depending on the noise level ξ. On the other hand when the noise term behaves in time like the fractional Brownian motion with index H∈(1∕2,1), We also derive explicit bounds leading to a well-known intermittency property.