# American Institute of Mathematical Sciences

July  2008, 20(3): 459-509. doi: 10.3934/dcds.2008.20.459

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

 1 Department of Mechanics and Mathematics, Kharkov National University, Kharkov, 61077 2 Department of Mathematics, University of Virginia, Charlottesville, VA 22903 3 Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588, United States

Received  January 2007 Revised  August 2007 Published  December 2007

This article addresses long-term behavior of solutions to a semilinear damped wave equation with a critical source term. A distinctive feature of the model is the geometrically constrained dissipation: it only affects a small subset of the domain adjacent to a connected portion of the boundary. The main result of the paper provides an affirmative answer to the open question whether global attractors for a wave equation with critical source and geometrically constrained damping are smooth and finite-dimensional. A positive answer to the same question in the case of subcritical sources was given in [9]. However, critical exponent of the source term combined with weak geometrically restricted dissipation constitutes the major new difficulty of the problem. To overcome this issue we develop a new version of Carleman's estimates and apply them in the context of recent results [12] on fractal dimension of global attractors.
Citation: Igor Chueshov, Irena Lasiecka, Daniel Toundykov. Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent. Discrete & Continuous Dynamical Systems, 2008, 20 (3) : 459-509. doi: 10.3934/dcds.2008.20.459
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