Uniform stabilization for elastic waves system with highly nonlinear localized dissipation (Q1395908)

The authors study decay properties of the solutions for the following initial-boundary value problem related with the system of elastic waves with a localized nonlinear dissipative term \[ \begin{cases} u_{tt}- b^2\Delta u- (a^2- b^2)\nabla\text{ div }u+\alpha u+ \rho(x,u_t)= 0,\\ u(x,0)= u_0(x),\;u_t(x,0)= u_1(x)\quad &\text{in }\Omega,\\ u(x,t)= 0\quad &\text{in }\partial\Omega\times \mathbb{R},\end{cases}\tag{1} \] where the medium \(\Omega\) is a bounded domain in \(\mathbb{R}^3\) with smooth boundary. Under some natural assumptions on \(b\), \(a\), \(u_0\), \(u_1\) the authors show that the solutions of (1) decay in an algebraic rate to zero.