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

Igor Chueshov; Irena Lasiecka; Daniel Toundykov

doi:10.3934/dcds.2008.20.459

Article Contents

2008, Volume 20, Issue 3: 459-509. Doi: 10.3934/dcds.2008.20.459

This issue Previous Article Elliptic PDE's in probability and geometry: Symmetry and regularity of solutions Next Article Toda system and interior clustering line concentration for a singularly perturbed Neumann problem in two dimensional domain

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

Received: December 31, 2006

Revised: July 31, 2007

Published: August 2008

Abstract Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 35B41; Secondary: 35B33, 35L70.

Citation:

Article Metrics

HTML views() PDF downloads(219) Cited by(0)

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

Abstract

Access History

Article Metrics

Access History

Other Articles By Authors

Catalog

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

Abstract

Access History

SHARE

Article Metrics

Access History

Other Articles By Authors

Catalog

Export File

Citation

Format

Content