Dynamical behaviors of solutions to nonlinear wave equations with vanishing local damping and Wentzell boundary conditions

Abstract

We are concerned with dynamical behaviors of solutions to nonlinear damped wave equations with nonlinear dampings and force terms, and subject to Wentzell boundary conditions which can be used to describe, for instance, the boundary behavior of a vibrating elastic body (resp. membrane) coated (resp. edged) with a thin layer (resp. coil) of high rigidity. Here the internal dampings are only assumed to be locally distributed and, especially, may disappear gradually over time. We find a new and effective method to overcome all the difficulties caused by the interplay of vanishing localized dampings, Wentzell boundary conditions, as well as nonlinear force terms. Ideal uniform decay rates of solution energies are obtained in terms of the exponents associated with the time-varying damping. Our result shows that the dynamical behavior of solutions is clearly stable without any bifurcation and chaos. To illustrate our theoretical results, we provide some numerical simulations.