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

TL;DR

This paper investigates the behavior of solutions to fractional stochastic heat equations driven by space-time colored noise in bounded domains, establishing bounds and growth properties depending on noise correlation and intensity.

Contribution

It provides new bounds and growth rate analysis for solutions under different temporal correlation structures of the noise.

Findings

Solution growth changes with noise level when noise is temporally uncorrelated.

Explicit bounds demonstrate intermittency when noise has fractional Brownian motion characteristics.

Results extend understanding of stochastic heat equations with colored noise in bounded domains.

Abstract

We consider the fractional stochastic heat type equation \begin{align*} \frac{\partial}{\partial t} u_t(x)=-(-\Delta)^{\alpha/2}u_t(x)+\xi\sigma(u_t(x))\dot{F}(t,x),\ \ \ x\in D, \ \ t>0, \end{align*} with nonnegative bounded initial condition, where $\alpha\in (0,2]$, $\xi>0$ is the noise level, $\sigma:\mathbb{R}\rightarrow\mathbb{R}$ is a globally Lipschitz function satisfying some growth conditions and the noise term 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 $\mathbb{R}^d$. 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 $\xi$. On the other hand when the noise term behaves in time like the fractional Brownian motion with index $H\in (1/2,1)$, We also derive explicit bounds leading to a well-known intermittency property.