  Consider the semilinear heat equation $\partial_t u = \partial^2_x u +
\lambda\sigma(u)\xi$ on the interval $[0\,,1]$ with Dirichlet zero boundary
condition and a nice non-random initial function, where the forcing $\xi$ is
space-time white noise and $\lambda>0$ denotes the level of the noise. We show
that, when the solution is intermittent [that is, when
$\inf_z|\sigma(z)/z|>0$], the expected $L^2$-energy of the solution grows at
least as $\exp\{c\lambda^2\}$ and at most as $\exp\{c\lambda^4\}$ as
$\lambda\to\infty$. In the case that the Dirichlet boundary condition is
replaced by a Neumann boundary condition, we prove that the $L^2$-energy of the
solution is in fact of sharp exponential order $\exp\{c\lambda^4\}$. We show
also that, for a large family of one-dimensional randomly-forced wave
equations, the energy of the solution grows as $\exp\{c\lambda\}$ as
$\lambda\to\infty$. Thus, we observe the surprising result that the stochastic
wave equation is, quite typically, significantly less noise-excitable than its
parabolic counterparts.
