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Non-linear noise excitation and intermittency under high disorder. (English) Zbl 1321.60136

The authors consider the semi-linear heat equation \[ \partial_tu = \partial_x^2u+\lambda\sigma(u)\xi \] on the interval \([0,L]\), with vanishing Dirichlet boundary condition and regular initial function, where \(\xi\) denotes space-time white noise. They show that if the solution is intermittent (i.e. \(\inf_z|\sigma(z)/z|>0\)), the rate of growth of the expected \(L^2\)-energy of the solution has, respectively, lower and upper bounds of the form \(\exp\{c\lambda^2\}\) and \(\exp\{c\lambda^4\}\) as \(\lambda\to\infty\). They further show that if the Dirichlet boundary condition is replaced by a Neumann boundary condition, then the \(L^2\)-energy of the solution is of sharp exponential order \(\exp\{c\lambda^4\}\) and that for a large family of one-dimensional randomly forced wave equations, the energy of the solution grows at the rate of \(\exp\{c\lambda\}\) as \(\lambda\to\infty\). This yields the surprising conclusion that the stochastic wave equation is typically less noise-excitable than its parabolic counterparts.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
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