Non-linear noise excitation and intermittency under high disorder∗

Consider the semilinear heat equation ∂tu = ∂ xu + λσ(u)ξ on the interval [0 , 1] with Dirichlet zero boundary condition and a nice nonrandom initial function, where the forcing ξ is space-time white noise and λ > 0 denotes the level of the noise. We show that, when the solution is intermittent [that is, when infz |σ(z)/z| > 0], the expected L-energy of the solution grows at least as exp{cλ} and at most as exp{cλ} as λ→∞. In the case that the Dirichlet boundary condition is replaced by a Neumann boundary condition, we prove that the Lenergy of the solution is in fact of sharp exponential order exp{cλ}. We show also that, for a large family of one-dimensional randomlyforced wave equations, the energy of the solution grows as exp{cλ} as λ → ∞. Thus, we observe the surprising result that the stochastic wave equation is, quite typically, significantly less noise-excitable than its parabolic counterparts.