On the behaviour of stochastic heat equations on bounded domains

Abstract

Consider the following equation $${\partial }_t{u}_t(x)=\frac{1}{2}{\partial }_{xx}{u}_t(x)+\lambda \sigma (u_t(x))\dot{W}(t,x)$$ on an interval. Under Dirichlet boundary condition, we show that in the long run, the second moment of the solution grows exponentially fast if $\lambda$ is large enough. But if $\lambda$ is small, then the second moment eventually decays exponentially. If we replace the Dirichlet boundary condition by the Neumann one, then the second moment grows exponentially fast no matter what $\lambda$ is. We also provide various extensions.

References in corpus (1)

• Strong invariance and noise-comparison principles for some parabolic stochastic PDEs

Cited by in corpus (1)

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