  We consider a stochastic heat equation of the type, $\partial_t u =
\partial^2_x u + \sigma(u)\dot{W}$ on $(0\,,\infty)\times[-1\,,1]$ with
periodic boundary conditions and on-degenerate positive initial data, where
$\sigma:\mathbb{R} \to\mathbb{R}$ is a non-random Lipschitz continuous function
and $\dot{W}$ denotes space-time white noise. If additionally $\sigma(0)=0$
then the solution is known to be strictly positive; see Mueller '91. In that
case, we prove that the oscillation of the logarithm of the solution decays
sublinearly as time tends to infinity. Among other things, it follows that,
with probability one, all limit points of $t^{-1}\, \sup_{x\in[-1,1]}\, \log
u(t\,,x)$ and $t^{-1}\, \inf_{x\in[-1,1]}\, \log u(t\,,x)$ must coincide. As a
consequence of this fact, we prove that, when $\sigma$ is linear, there is a.s.
only one such limit point and hence the entire path decays almost surely at an
exponential rate.
