Existence of invariant manifolds for coupled parabolic and hyperbolic stochastic partial differential equations

Abstract

An abstract system of coupled nonlinear parabolic–hyperbolic partial differential equations subjected to additive white noise is considered. The system models temperature dependent or heat generating wave phenomena in a continuum random medium. Under suitable conditions, the existence of an exponentially attracting random invariant manifold for the coupled system is proved, and as a consequence, the system can be reduced to a single stochastic hyperbolic equation with a modified nonlinear term. Finally, it is also proved that this random manifold converges to its deterministic counterpart when the intensity of noise tends to zero.