Global Attractor for a Wave Equation with Nonlinear Localized Boundary Damping and a Source Term of Critical Exponent

The paper addresses long-term behavior of solutions to a damped wave equation with a critical source term. The dissipative frictional feedback is restricted to a subset of the boundary of the domain. This paper derives inverse observability estimates which extend the results of Chueshov et al. (Disc Contin Dyn Syst 20:459–509, 2008) to systems with boundary dissipation. In particular, we show that a hyperbolic flow under a critical source and geometrically constrained boundary damping approaches a smooth finite-dimensional global attractor. A similar result for subcritical sources was given in Chueshov et al. (Commun Part Diff Eq 29:1847–1876, 2004). However, the criticality of the source term in conjunction with geometrically restricted dissipation constitutes the major new difficulty of the problem. To overcome this issue we develop a special version of Carleman’s estimates and apply them in the context of abstract results on dissipative dynamical systems. In contrast with the localized interior damping (Chueshov et al. Disc Contin Dyn Syst 20:459–509, 2008), the analysis of a boundary feedback requires a more careful treatment of the trace terms and special tangential estimates based on microlocal analysis.