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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

Igor Chueshov 1, , Irena Lasiecka 2, and  Daniel Toundykov 3,

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.
Keywords: localized damping, critical exponents., attractor, source, fractal dimension, wave equation, nonlinear damping.
Mathematics Subject Classification: Primary: 35B41; Secondary: 35B33, 35L7.
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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