Stabilization of the wave equation with localized nonlinear damping (Q1265110)

The author considers an initial value problem for a nonlinearly damped wave equation \[ \ddot y-\Delta y+ ag(\dot y)= 0\tag{1} \] in a bounded domain \(\Omega\subseteq \mathbb{R}^N\), with the boundary condition \(y= 0\) on \(\Gamma= \partial\Omega\). Here \(g(x)\) is a continuous nondecreasing function satisfying \(g(0)= 0\), and \(a(x)\) is a nonnegative bounded function such that \(a(x)\geq a_0>0\) in a neighbourhood of \(\Gamma\). Under certain conditions on the growth of \(g(x)\), several time-decay estimates are established for the energy functional associated with (1): \[ E= {1\over 2} \int_\Omega \{\dot y^2+(\nabla y)^2\}dx. \] The method of proof is direct and is based on the multiplier technique, on some integral inequalities due to Haraux and Komornik, and on a judicious idea of Conrad and Rao.