On a stochastic nonlinear equation in one-dimensional viscoelasticity (Q2759057)

An initial boundary-value problem associated with one-dimensional viscoelasticity of rate type with random force of the following type: \( u_{tt} =\eta(u_{x},u_{tx}) + \xi + f \), where \( u(t,x) \) is the displacement, \( \eta \) is the stress function which depends on gradient \( u_{x} \) and its derivative \( u_{tx} \), \( \xi \) is a random perturbation depending on \( u_{tx} \) and \( f \) represents a given random force, is considered. The author points out references where special cases of the equation under consideration (e.g. deterministic equation, semilinear stochastic evolution equation) have been earlier investigated. The main difficulty in handling the nonlinearity in the equation is the lack of compactness of approximate solutions which ensure their strong convergence. The author presents a new direct approach, based on a scheme of Galerkin approximation and a new method of constructing a pathwise solution, for obtaining a solution to the equation. The method is expected to be applicable to other nonlinear stochastic PDEs.