Uniform decay rates of coupled anisotropic elastodynamic/Maxwell equations with nonlinear damping

Abstract

This work is devoted to study the asymptotic behavior of the total energy associated with a coupled system of anisotropic hyperbolic models: the elastodynamic equations and Maxwell's system in the exterior of a bounded body in R 3 . Our main result says that in the presence of nonlinear damping, a unique solution of small initial data exists globally in time and the total energy as well as higher order energies decay at a uniform rate as t ! þl.

Stability properties of dissipative evolution equations with nonautonomous and nonlinear damping

“…Stability of evolution equations of hyperbolic type with linear or nonlinear autonomous feedbacks has been the object of many works. Let us quote the stability of the wave equation [29,30,32,33,35,40,64], of the elastodynamic system [1,8,18,19,20,22,36,61], of the Petrovsky system [17,33,34], of Mawxell's system [5,15,31,54,59] or combination of them [13,25,53], see also the references cited in the aforementioned works. On the contrary the case of nonautonomous damping is less considered in the literature, let us quote [14,23,45,46,47,49,50,51,60] for the wave equation and [6,7] for the Lamé systems.…”