Moment bounds for a class of fractional stochastic heat equations

TL;DR

This paper investigates the growth of moments in fractional stochastic heat equations with noise, establishing exponential growth of the second moment and addressing an open problem in the field.

Contribution

It provides new moment bounds for fractional stochastic heat equations, extending existing results and solving an open problem regarding moment growth.

Findings

Second moment of solutions grows exponentially over time.

Established bounds extend previous theoretical results.

Addresses an open problem in stochastic PDEs literature.

Abstract

We consider fractional stochastic heat equations of the form $\frac{\partial u_t(x)}{\partial t} = -(-\Delta)^{\alpha/2} u_t(x)+\lambda \sigma (u_t(x)) \dot F(t,\, x)$. Here $\dot F$ denotes the noise term. Under suitable assumptions, we show that the second moment of the solution grows exponentially with time. In particular, this answers an open problem in \cite{CoKh}. Along the way, we prove a number of other interesting properties which extend and complement results in \cite{foonjose}, \cite{Khoshnevisan:2013aa} and \cite{Khoshnevisan:2013ab}.