Uniform decay rates and attractors for evolution PDE's with boundary dissipation (Q1898789)

The main goal of the paper is to study the long time behaviour of solutions to some evolutionary equations with dissipative boundary conditions. A typical example of such a problem is the semilinear wave equation of the form \(u_{tt} - \Delta u + f(u) = 0\) in \(\Omega \subset \mathbb{R}^N\) complemented by the boundary conditions \({\partial u / \partial \nu} + u_t = 0\) on \(\partial \Omega\). Under various growth conditions concerning the function \(f\), the existence of a compact attractor or, if \(f\) is coercive, the convergence to zero of solutions for large times is established. The case of the so-called critical exponent is included. A related problem for the Schrödinger equation is also discussed. The proof is based on an observability lemma for a general linear operator based on Carleman type estimates. Various possibilities of obtaining such a result and possible generalizations for specific equations are discussed.