Approximating Dynamics of a Singularly Perturbed Stochastic Wave Equation with a Random Dynamical Boundary Condition

Abstract

This work is concerned with a singularly perturbed stochastic nonlinear wave equation with a random dynamical boundary condition. A splitting is used to establish the approximating equation of the system for a sufficiently small singular perturbation parameter. The approximating equation is a stochastic parabolic equation when the power exponent of the singular perturbation parameter is in $[1/2, 1)$ but is a deterministic wave equation when the power exponent is in $(1, +\infty)$. Moreover, if the power exponent of a singular perturbation parameter is bigger than or equal to $1/2$, the same limiting equation of the system is derived in the sense of distribution, as the perturbation parameter tends to zero. This limiting equation is a deterministic parabolic equation.