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
Full-text PDF Free Access
Abstract |
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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 $A$ 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.
References
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- Francesca Bucci and Irena Lasiecka, Exponential decay rates for structural acoustic model with an overdamping on the interface and boundary layer dissipation, Appl. Anal. 81 (2002), no. 4, 977–999. MR 1930118, DOI https://doi.org/10.1080/0003681021000004555
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- Angelo Favini, Gisèle Ruiz Goldstein, Jerome A. Goldstein, and Silvia Romanelli, The heat equation with generalized Wentzell boundary condition, J. Evol. Equ. 2 (2002), no. 1, 1–19. MR 1890879, DOI https://doi.org/10.1007/s00028-002-8077-y
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- Ciprian G. Gal, Gisèle Ruiz Goldstein, and Jerome A. Goldstein, Oscillatory boundary conditions for acoustic wave equations, J. Evol. Equ. 3 (2003), no. 4, 623–635. Dedicated to Philippe Bénilan. MR 2058054, DOI https://doi.org/10.1007/s00028-003-0113-z
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- Gisèle Ruiz Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Differential Equations 11 (2006), no. 4, 457–480. MR 2215623
- E. Zuazua, On the overdamping phenomenon, unpublished communication (2001).
References
- J.T. Beale and S.I. Rosencrans, Acoustic boundary conditions, Bull. Amer. Math. Soc. 80 (1974), 1276–1278. MR 0348274 (50:772)
- J.T. Beale, Spectral properties of an acoustic boundary condition, Indiana University Math. J. 25 (1976), 895–917. MR 0408425 (53:12189)
- F. Bucci and I. Lasiecka, Exponential decay rates for structural acoustic model with an overdamping on the interface and boundary layer dissipation, Appl. Anal. 81 (2002), 977–999. MR 1930118 (2003i:93037)
- R. Datko, An example of an unstable neutral differential equation, Internat. J. Control 38 (1983), 263–267. MR 713318 (84k:34080)
- R. Datko and Y.C. You, Some second-order vibrating systems cannot tolerate small time delays in their damping, J. Optim. Theory Appl. 70 (1991), 521–537. MR 1124776 (92m:34132)
- R. Datko, J. Lagnese, and M.P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim. 24 (1986), 152–156. MR 818942 (87k:93074)
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Additional Information
Gisèle Ruiz Goldstein
Affiliation:
Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee 38152
MR Author ID:
333750
Email:
ggoldste@memphis.edu
Jerome A. Goldstein
Affiliation:
Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee 38152
MR Author ID:
74805
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
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