Analysis and discretization of semi-linear stochastic wave equations with cubic nonlinearity and additive space-time noise

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June 2008, 1(2): 353-363. doi: 10.3934/dcdss.2008.1.353

Analysis and discretization of semi-linear stochastic wave equations with cubic nonlinearity and additive space-time noise

1.	Southern Illinois University, Department of Mathematics, MC 4408, 1245 Lincoln Drive, Carbondale, IL 62901-7316

Received September 2006 Revised September 2007 Published March 2008

Abstract
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One-dimensional wave equations with cubic power law perturbed by Q-regular additive space-time random noise are considered. These models describe the displacement of nonlinear strings excited by state-independent random external forces. The presented analysis is based on the representation of its solution in form of Fourier-series expansions along the eigenfunctions of Laplace operator with continuous, Markovian, unique Fourier coefficients (the so-called commutative case). We shall discuss existence and uniqueness of Fourier solutions using energy-type methods based on the construction of Lyapunov-functions. Appropriate truncations and finite-dimensional approximations are presented while exploiting the explicit knowledge on eigenfunctions of related second order differential operators. Moreover, some nonstandard partial-implicit difference methods for their numerical integration are suggested in order to control its energy functional in a dynamically consistent fashion. The generalized energy $\cE$ (sum of kinetic, potential and damping energy) is governed by the linear relation $\E [\varepsilon(t)] = \E [\varepsilon(0)] + b^2 trace (Q) t / 2$ in time $t \ge 0$, where $b$ is the scalar intensity of noise and $Q$ its covariance operator.

Keywords: Fourier expansions., Lyapunov functionals, cubic nonlinearity, discretization of energy, additive noise, white noise, Semilinear stochastic wave equations, truncation technique, energy identity, partial-implicit numerical method, space-time noise.

Mathematics Subject Classification: Primary: 34F05, 60H10, 60H15; Secondary: 65C3.

Citation: Henri Schurz. Analysis and discretization of semi-linear stochastic wave equations with cubic nonlinearity and additive space-time noise. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 353-363. doi: 10.3934/dcdss.2008.1.353

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