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), 183199
MSC (2010):
Primary 35Q99, 35L99; Secondary 47D06, 35K10, 47N20
Published electronically:
August 28, 2012
MathSciNet review:
3075540
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We study the best possible energy decay rates for a class of linear secondorder 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.
 1.
J.
Thomas Beale and Steven
I. Rosencrans, Acoustic boundary conditions,
Bull. Amer. Math. Soc. 80 (1974), 1276–1278. MR 0348274
(50 #772), 10.1090/S000299041974137146
 2.
J.
Thomas Beale, Spectral properties of an acoustic boundary
condition, Indiana Univ. Math. J. 25 (1976),
no. 9, 895–917. MR 0408425
(53 #12189)
 3.
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
(2003i:93037), 10.1080/0003681021000004555
 4.
R.
Datko, An example of an unstable neutral differential
equation, Internat. J. Control 38 (1983), no. 1,
263–267. MR
713318 (84k:34080), 10.1080/00207178308933074
 5.
R.
Datko and Y.
C. You, Some secondorder vibrating systems cannot tolerate small
time delays in their damping, J. Optim. Theory Appl.
70 (1991), no. 3, 521–537. MR 1124776
(92m:34132), 10.1007/BF00941300
 6.
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), no. 1, 152–156. MR 818942
(87k:93074), 10.1137/0324007
 7.
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 (2003b:35089), 10.1007/s000280028077y
 8.
Angelo
Favini, Gisèle
Ruiz Goldstein, Jerome
A. Goldstein, and Silvia
Romanelli, The onedimensional wave equation with Wentzell boundary
conditions, Differential equations and control theory (Athens, OH,
2000) Lecture Notes in Pure and Appl. Math., vol. 225, Dekker, New
York, 2002, pp. 139–145. MR 1890560
(2003c:35108)
 9.
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
(2005g:35188), 10.1007/s000280030113z
 10.
Jerome
A. Goldstein, Semigroups of linear operators and applications,
Oxford Mathematical Monographs, The Clarendon Press, Oxford University
Press, New York, 1985. MR 790497
(87c:47056)
 11.
Michael
Reed and Barry
Simon, Methods of modern mathematical physics. II. Fourier
analysis, selfadjointness, Academic Press [Harcourt Brace Jovanovich,
Publishers], New YorkLondon, 1975. MR 0493420
(58 #12429b)
Michael
Reed and Barry
Simon, Methods of modern mathematical physics. III, Academic
Press [Harcourt Brace Jovanovich, Publishers], New YorkLondon, 1979.
Scattering theory. MR 529429
(80m:81085)
Michael
Reed and Barry
Simon, Methods of modern mathematical physics. IV. Analysis of
operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New
YorkLondon, 1978. MR 0493421
(58 #12429c)
 12.
Gisèle
Ruiz Goldstein, Derivation and physical interpretation of general
boundary conditions, Adv. Differential Equations 11
(2006), no. 4, 457–480. MR 2215623
(2006m:35130)
 13.
E. Zuazua, On the overdamping phenomenon, unpublished communication (2001).
 1.
 J.T. Beale and S.I. Rosencrans, Acoustic boundary conditions, Bull. Amer. Math. Soc. 80 (1974), 12761278. MR 0348274 (50:772)
 2.
 J.T. Beale, Spectral properties of an acoustic boundary condition, Indiana University Math. J. 25 (1976), 895917. MR 0408425 (53:12189)
 3.
 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), 977999. MR 1930118 (2003i:93037)
 4.
 R. Datko, An example of an unstable neutral differential equation, Internat. J. Control 38 (1983), 263267. MR 713318 (84k:34080)
 5.
 R. Datko and Y.C. You, Some secondorder vibrating systems cannot tolerate small time delays in their damping, J. Optim. Theory Appl. 70 (1991), 521537. MR 1124776 (92m:34132)
 6.
 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), 152156. MR 818942 (87k:93074)
 7.
 A. Favini, G.R. Goldstein, J.A. Goldstein and S. Romanelli, The heat equation with general Wentzell boundary conditions, J. Evol. Eq. 2 (2002), 119. MR 1890879 (2003b:35089)
 8.
 A. Favini, G.R. Goldstein, J.A. Goldstein and S. Romanelli, The onedimensional wave equation with Wentzell boundary conditions, Differential Equations and Control Theory (ed. by S. Aizicovici and N. Pavel), M. Dekker, New York, 2002, 139145. MR 1890560 (2003c:35108)
 9.
 C. Gal, G.R. Goldstein and J.A. Goldstein, Oscillatory boundary conditions for acoustic wave equations, J. Evol. Eqns., 3 (2003), 623635. MR 2058054 (2005g:35188)
 10.
 J.A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford Univ. Press, N.Y., 1985. MR 790497 (87c:47056)
 11.
 M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. II, III and IV, Academic Press, New York  London 1975, 1978, 1979. MR 0493420 (58:12429b); MR 0529429 (80m:81085); MR 0493421 (58:12429c)
 12.
 G.R. Goldstein, Derivation and physical interpretation of general boundary conditions, Advances in Diff. Equations, 11 (2006), 457480. MR 2215623 (2006m:35130)
 13.
 E. Zuazua, On the overdamping phenomenon, unpublished communication (2001).
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2010):
35Q99,
35L99,
47D06,
35K10,
47N20
Retrieve articles in all journals
with MSC (2010):
35Q99,
35L99,
47D06,
35K10,
47N20
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 25651070, 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:
http://dx.doi.org/10.1090/S0033569X2012012823
PII:
S 0033569X(2012)012823
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. 301134/20090) and Project Universal (Proc. 47296/20083) 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
