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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 thestructure 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 compactsupport, whereas with probability one,$\limsup_{|x|\to\infty}u_t(x)/({\log}|x|)^{1/6}>0$ when $u_0$ is boundeduniformly away from zero. This sensitivity to the initial data of thestochastic heat equation is a way to state that the solution to the stochasticheat equation is chaotic at fixed times, well before the onset ofintermittency.

On the chaotic character of the stochastic heat equation, before the onset of intermitttency