  This paper studies the long time stability of both stochastic heat equations
on a bounded domain driven by a correlated noise and their approximations. It
is popular for researchers to prove the intermittency of the solution which
means that the moments of solution to stochastic heat equation usually grow
exponentially to infinite and this hints that the solution to stochastic heat
equation is generally not stable in long time. However, quite surprisingly in
this paper we show that when the domain is bounded and when the noise is not
singular in spatial variables, the system can be long time stable and we also
prove that we can approximate the solution by its finite dimensional spectral
approximation which is also long time stable. The idea is to use eigenfunction
expansion of the Laplacian on bounded domain. We also present numerical
experiments which are consistent with our theoretical results.
