Effects of the noise level on stochastic fractional heat equations

TL;DR

This paper studies how different noise levels affect the behavior of solutions to stochastic fractional heat equations, showing exponential growth bounds, stability for small noise, and the noise excitation index as noise increases.

Contribution

It provides new insights into the growth, stability, and excitation index of solutions to stochastic fractional heat equations under varying noise intensities.

Findings

Exponential growth bound of the p-th moment of the solution's supremum.

Exponential stability of the solution for small noise levels.

Determination of the noise excitation index as noise intensity tends to infinity.

Abstract

We consider the stochastic fractional heat equation $\partial_{t}u=\triangle^{\alpha/2}u+\lambda\sigma(u)\dot{w}$ on $[0,L]$ with Dirichlet boundary conditions, where $\dot{w}$ denotes the space-time white noise. For any $\lambda>0$, we prove that the $p$th moment of $\sup_{x\in [0,L]}|u(t,x)|$ grows at most exponentially. Moreover, we prove that the $p$th moment of $\sup_{x\in [0,L]}|u(t,x)|$ is exponentially stable if $\lambda$ is small. At last, We obtain the noise excitation index of $p$th energy of $u(t,x)$ as $\lambda\rightarrow \infty$.