Decay of solutions of the wave equation with arbitrary localized nonlinear damping (Q1775526)

The author studies initial boundary value problem to the equation \(u_{tt}-\triangle u+(1+t)^{\theta }a(x)g(u_t)=0,\) \(x\in \Omega\), \(t\in (0,\infty ),\) with given initial data and Dirichlet conditions on the boundary, where \(-1<\theta \leq 0\) and \(\Omega \) is a smooth Riemannian compact manifold in \(\mathbb R^n.\) Under some assumptions on \(a(x), g(s),\) he proves the logarithmic decay in time of the unique classical solution of the problem.