Decay of solutions of the wave equation with a local nonlinear dissipation
We derive precise decay estimates of the energy for the solutions to the initial-boundary value problem for the wave equation with a nonlinear dissipative term \[ u_{tt}- \Delta u+ \rho(x, u_t)= 0\quad \text{in } \Omega\times [0, \infty) \] with the boundary condition \(u|_{\partial\Omega}= 0\), where \(\rho(x, v)\) is a function like \(\rho(x, v)= a(x)|v|^r v\), \(- 1< r\), with \(a(x)\) supported only on a neighbourhood of some part of the boundary \(\partial\Omega\).
scientific article; zbMATH DE number 931967