Stability to weak dissipative Bresse system

In this paper we study the Bresse system with frictional dissipation working only on the angle displacement. Our main result is to prove that this dissipative mechanism is enough to stabilize exponentially the whole system provided the velocities of waves propagations are the same. This result is si...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2011-02, Vol.374 (2), p.481-498
Main Authors: Alabau Boussouira, Fatiha, Muñoz Rivera, Jaime E., Almeida Júnior, Dilberto da S.
Format: Article
Language:English
Subjects:
Analysis of PDEs
Bresse system
Exact sciences and technology
Finite difference
Global analysis, analysis on manifolds
Group theory
Group theory and generalizations
Lack of exponential decay
Mathematical analysis
Mathematics
Optimization and Control
Sciences and techniques of general use
Semigroups
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
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Summary:In this paper we study the Bresse system with frictional dissipation working only on the angle displacement. Our main result is to prove that this dissipative mechanism is enough to stabilize exponentially the whole system provided the velocities of waves propagations are the same. This result is significative only from the mathematical point of view since in practice the velocities of waves propagations are always different. In that direction we show that when the velocities are not the same, the system is not exponentially stable and we prove that the solution in this case goes to zero polynomially, with rates that can be improved by taking more regular initial data. Finally, we give some numerical result to verify our analytical results.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2010.07.046