Stochastic wave equations with cubic nonlinearity and Q-regular additive noise in $\mathbb{R}^2$

Pages: 1299 - 1308, Issue Special, September 2011

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Henri Schurz - Southern Illinois University, Department of Mathematics, MC 4408, 1245 Lincoln Drive, Carbondale, IL 62901-7316, United States (email)

Abstract: Semi-linear wave equations on rectangular domains in $\mathbb{R}^2$ (vibrating plates) with certain cubic quasi-nonlinearities and perturbed by a Q-regular space-time white noise are considered analytically. These models as 2nd order SPDEs (stochastic partial differential equations) with non-random Dirichlet- type boundary conditions describe the displacement of noisy vibrations of rectangular plates as met in engineering. We discuss their analysis by the eigen- function approach allowing us to truncate the infinite-dimensional stochastic systems (i.e. the SDEs of Fourier coefficients related to semilinear SPDEs), to control its energy, existence, uniqueness, continuity and stability. A conservation law for at most linearly growing expected energy is established in terms of system-parameters.

Keywords:  Semilinear wave equations; SPDEs; cubic nonlinearity; Q-regular space-time noise; Gaussian noise; Wiener process; Lyapunov functions; total energy functional; Fourier solutions; approximating Fourier coecients; conservation laws; trace formula.
Mathematics Subject Classification:  Primary: 34F05, 37H10, 60H10, 60H30; Secondary: 65C30

Received: July 2010;      Revised: March 2011;      Published: October 2011.