On the behaviour of stochastic heat equations on bounded domains

Consider the following equation $$partial_t u_t(x)=frac{1}{2}partial _{xx}u_t(x)+lambda sigma(u_t(x))dot{W}(t,,x)$$ 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.