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. 
Mohammud Foondun and Eulalia Nualart
math.PR