On the overdamping phenomenon: A general result and applications

Authors:
Gisèle Ruiz Goldstein, Jerome A. Goldstein and Gustavo Perla Menzala

Journal:
Quart. Appl. Math. **71** (2013), 183-199

MSC (2010):
Primary 35Q99, 35L99; Secondary 47D06, 35K10, 47N20

DOI:
https://doi.org/10.1090/S0033-569X-2012-01282-3

Published electronically:
August 28, 2012

MathSciNet review:
3075540

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Abstract | References | Similar Articles | Additional Information

Abstract: We study the best possible energy decay rates for a class of linear second-order dissipative evolution equations in a Hilbert space. The models we consider are generated by a positive selfadjoint operator having a bounded inverse. Our discussion applies to important examples such as the classical wave equation, the dynamical wave equation with Wentzell boundary conditions and many others.

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Additional Information

**Gisèle Ruiz Goldstein**

Affiliation:
Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee 38152

Email:
ggoldste@memphis.edu

**Jerome A. Goldstein**

Affiliation:
Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee 38152

Email:
jgoldste@memphis.edu

**Gustavo Perla Menzala**

Affiliation:
National Laboratory of Scientific Computation (LNCC/MCT), Ave. Getulio Vargas 333, Quitandinha, Petropolis, RJ, CEP 25651-070, Brasil and Federal University of Rio de Janeiro, Institute of Mathematics, P.O. Box 68530, Rio de Janeiro, RJ, Brazil

Email:
perla@lncc.br

DOI:
https://doi.org/10.1090/S0033-569X-2012-01282-3

Keywords:
Overdamping phenomenon,
the spectral theorem,
classical wave equations,
Wentzell boundary conditions

Received by editor(s):
May 26, 2011

Published electronically:
August 28, 2012

Additional Notes:
Part of this work was done during the visit of GG and JG to the National Laboratory of Scientific Computation (Brasil) during July 2008. The Goldsteins are most grateful for the gracious hospitality they received from G. Perla Menzala and Carlos Moura.

The third author was partially supported by a Research Grant of CNPq (Proc. 301134/2009-0) and Project Universal (Proc. 47296/2008-3) from the Brazilian Government. The third author would like to express his gratitude for such important support. He is also very thankful to Prof. E. Zuazua. Several years ago he discussed with him the overdamping phenomenon ([13]) and its implications.

Article copyright:
© Copyright 2012
Brown University