  We consider nonlinear parabolic SPDEs of the form $\partial_t u=\sL u +
\sigma(u)\dot w$, where $\dot w$ denotes space-time white noise,
$\sigma:\R\to\R$ is [globally] Lipschitz continuous, and $\sL$ is the
$L^2$-generator of a L\'evy process. We present precise criteria for existence
as well as uniqueness of solutions. More significantly, we prove that these
solutions grow in time with at most a precise exponential rate. We establish
also that when $\sigma$ is globally Lipschitz and asymptotically sublinear, the
solution to the nonlinear heat equation is ``weakly intermittent,'' provided
that the symmetrization of $\sL$ is recurrent and the initial data is
sufficiently large.
  Among other things, our results lead to general formulas for the upper
second-moment Liapounov exponent of the parabolic Anderson model for $\sL$ in
dimension $(1+1)$. When $\sL=\kappa\partial_{xx}$ for $\kappa>0$, these
formulas agree with the earlier results of statistical physics
\cite{Kardar,KrugSpohn,LL63}, and also probability theory \cite{BC,CM94} in the
two exactly-solvable cases where $u_0=\delta_0$ and $u_0\equiv 1$.
