Non-linear noise excitation and intermittency under high disorder

Consider the semilinear heat equation $\partial_t u = \partial^2_x u + \lambda\sigma(u)\xi$ontheinterval$[0,,1]$ 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 ${\inf }_z\mathrm{\mid }\sigma (z)\mathrm{/}z\mathrm{\mid }>0$], 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 $\lambda \to \mathrm{\infty }$. 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 $\lambda \to \mathrm{\infty }$. Thus, we observe the surprising result that the stochastic wave equation is, quite typically, significantly less noise-excitable than its parabolic counterparts.