  We consider a nonlinear stochastic heat equation
$\partial_tu=\frac{1}{2}\partial_{xx}u+\sigma(u)\partial_{xt}W$, where
$\partial_{xt}W$ denotes space-time white noise and $\sigma:\mathbf {R}\to
\mathbf {R}$ is Lipschitz continuous. We establish that, at every fixed time
$t>0$, the global behavior of the solution depends in a critical manner on the
structure of the initial function $u_0$: under suitable conditions on $u_0$ and
$\sigma$, $\sup_{x\in \mathbf {R}}u_t(x)$ is a.s. finite when $u_0$ has compact
support, whereas with probability one,
$\limsup_{|x|\to\infty}u_t(x)/({\log}|x|)^{1/6}>0$ when $u_0$ is bounded
uniformly away from zero. This sensitivity to the initial data of the
stochastic heat equation is a way to state that the solution to the stochastic
heat equation is chaotic at fixed times, well before the onset of
intermittency.
