Effects of the noise level on stochastic fractional heat equations

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 $\lambda\rightarrow \infty$.