Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent

Abstract

This article addresses long-term behavior of solutions to asemilinear damped wave equation with a critical source term. Adistinctive feature of the model is the geometrically constraineddissipation: it only affects a small subset of the domainadjacent to a connected portion of the boundary. The main result ofthe paper provides an affirmative answer to the open questionwhether global attractors for a wave equation with criticalsource and geometrically constrained damping are smooth andfinite-dimensional. A positive answer to the same question in thecase of subcritical sources was given in[9]. However, critical exponent of the sourceterm combined with weak geometrically restricted dissipationconstitutes the major new difficulty of the problem. To overcomethis issue we develop a new version of Carleman's estimates andapply them in the context of recent results [12]on fractal dimension of global attractors.