Wave equation on a bounded domain with boundary dissipation: An operator approach (Q1262459)

The wave equation on a bounded domain, whose boundary \(\Gamma\) consists of two parts \(\Gamma_ 0\) and \(\Gamma_ 1\) \((\Gamma_ 1\neq \emptyset\) and relatively open in \(\Gamma)\) is studied with boundary dissipation as an abstract evolution problem in a suitable functional space. This space incorporates the boundary action. By means of the semigroup techniques and spectral properties, strong stabilization is proved. Uniform stabilization is proved with a multiplier approach both in the case \(\Gamma_ 0\neq \emptyset\) and \(\Gamma_ 0=\emptyset.\) An operator approach with a slightly modified version of a Lagnese theorem is used for the uniform stabilization proof.