Parabolic singular limit of a wave equation with localized interior damping (Q2732615)

The paper deals with the initial boundary value problem for the linear damped wave equation \( \epsilon u^{\epsilon}_{tt}+\chi _{\omega}u^{\epsilon}_{t}-\triangle u^{\epsilon}+\lambda u^{\epsilon} =f^{\epsilon}(x,t) \) in \(\Omega \times (0,T)\) with homogeneous Dirichlet conditions on the boundary of \(\Omega \) and some initial data, where \(\chi _{\omega}\) denotes the characteristic function of the open set \(\omega \subset \Omega \). The authors study the behavior of the solution \(u^{\epsilon} (x,t)\) as \(\epsilon\rightarrow 0,\) namely its convergence to the solution of the limiting parabolic-elliptic problem.