On the behaviour of stochastic heat equations on bounded domains

Consider the following equation $\partial_t u_t(x)=\frac{1}{2}\partial _{xx}u_t(x)+λσ(u_t(x))\dot{W}(t,\,x)$ on an interval. Under Dirichlet boundary condition, we show that in the long run, the second moment of the solution grows exponentially fast if $λ$ is large enough. But if $λ$ 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 $λ$ is. We also provide various extensions.

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