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

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}.