Weak asymptotic decay for a wave equation with gradient dependent damping (Q2730789)

The author proves weak asymptotic stabilization in \(H^{1}_{0}(\Omega)\times L^{2}(\Omega)\) of all global solutions of the Dirichlet problem to the equation \(u_{tt}-\triangle u=-a(x)\beta (u_{t},\nabla u)\) under weak assumptions on the damping \(\beta .\) The result is extended from the operator \(-\triangle \) to a linear coercive selfadjoint operator with compact resolvent. The same results are proved in the case of the wave equation with a boundary feedback \(\frac{\partial u}{\partial n}=-a(x)q(u_{t})\) on a part of the boundary \([0,\infty)\times \partial \Omega.\)