Non-linear noise excitation and intermittency under high disorder

Abstract:

Consider the semilinear heat equation $\partial _t u = \partial ^2_x u + \lambda \sigma (u)\xi$ on the interval $[0 ,L]$ 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 on $\mathbf {R}$, 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.