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

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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

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

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