Stabilization of the Euler–Bernoulli plate with variable coefficients by nonlinear internal feedback

Abstract

We consider the energy decay for solutions of the Euler–Bernoulli plate equation with variable coefficients where a nonlinear internal feedback acts in a suitable subregion of the domain. The Riemannian geometric method is used to deal with variable coefficient problems. When the feedback region has a structure similar to that for the wave equation with constant coefficients, we establish the stabilization of the system in the case of fixed boundary conditions. Several energy decay rates are established according to various growth restrictions on the nonlinear feedback near the origin and at infinity. We further show that, unlike for the case of constant coefficients, choices of such feedback regions depend not only on the type of boundary conditions but also on the curvature of a Riemannian metric, based on the coefficients of the system.