  Consider the stochastic heat equation
$\partial_tu=\mathscr{L}u+\lambda\sigma(u)\xi$, where $\mathscr{L}$ denotes the
generator of a L\'{e}vy process on a locally compact Hausdorff Abelian group
$G$, $\sigma:\mathbf{R}\to\mathbf{R}$ is Lipschitz continuous, $\lambda\gg1$ is
a large parameter, and $\xi$ denotes space-time white noise on
$\mathbf{R}_+\times G$. The main result of this paper contains a near-dichotomy
for the (expected squared) energy $\mathrm{E}(\|u_t\|_{L^2(G)}^2)$ of the
solution. Roughly speaking, that dichotomy says that, in all known cases where
$u$ is intermittent, the energy of the solution behaves generically as
$\exp\{\operatorname {const}\cdot\,\lambda^2\}$ when $G$ is discrete and
$\ge\exp\{\operatorname {const}\cdot\,\lambda^4\}$ when $G$ is connected.
