Some effects of the noise intensity upon non-linear stochastic heat equations on $[0,1]$

TL;DR

This paper studies how different levels of noise intensity affect the behavior of solutions to the stochastic heat equation on [0,1], revealing stability for small noise and exponential growth for large noise.

Contribution

It introduces a unified method to analyze the lower bounds of solution behavior under varying noise intensities, addressing previously difficult problems.

Findings

Small noise leads to exponential stability of the solution's supremum moments.

Large noise causes at least exponential growth in the solution's moments.

The noise excitation of the p-th energy approaches 4 as noise intensity increases.

Abstract

Various effects of the noise intensity upon the solution $u(t,x)$ of the stochastic heat equation with Dirichlet boundary conditions on $[0,1]$ are investigated. We show that for small noise intensity, the $p$-th moment of $\sup_{x \in [0,1]} |u(t,x)|$ is exponentially stable, however, for large one, it grows at least exponentially. We also prove that the noise excitation of the $p$-th energy of $u(t,x)$ is $4$, as the noise intensity goes to infinity. We formulate a common method to investigate the lower bounds of the above two different behaviors for large noise intensity, which are hard parts in \cite{FoJo-14}, \cite{FoNu} and \cite{KhKi-15}.