Uniform decay rates of solutions to a system of coupled PDEs with nonlinear internal dissipation

Abstract

We consider a coupled system of partial differential equations (PDEs) of hyperbolic/parabolic type, which is a generalization of an established structural acoustic model. A peculiar feature of this model is a high damping term in the boundary condition of the wave component which accounts for lack of uniform stability of the overall system, even in the presence of viscous damping on the entire domain. It has been shown recently that by introducing a comparable static damping in the boundary condition of the wave component, the corresponding feedback system is uniformly stable. In this paper we study the stability properties of the coupled system when the internal damping is subject to nonlinear effects. We provide two main different stability results, which describe decay rates of the solutions to the coupled PDE system via the solution to an appropriate nonlinear ordinary differential equation. Our analysis allows saturation of the nonlinear dissipation term. In this significant case we obtain as well uniform decay estimates of the underlying energy, provided that initial data are measured with a slightly stronger topology. Even though the prime concern of this paper is to deal with coupled structures and the overdamping phenomenon, these results yield, as well, new results on uniform stabilization of a single wave equation.