Slow decay in linear thermoelasticity
Author:
Herbert Koch
Journal:
Quart. Appl. Math. 58 (2000), 601-612
MSC:
Primary 74F05; Secondary 35B35, 35B40, 35Q72, 74H40
DOI:
https://doi.org/10.1090/qam/1788420
MathSciNet review:
MR1788420
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: Energy estimates show that linearized thermoelasticity defines a contraction semigroup on a Hilbert space. We show that under a geometric condition this contraction is not strict, or, more precisely, the norm of the semigroup is 1 for all $t \ge 0$. Convex domains always satisfy the geometric condition.
- Daniel B. Henry, Anisio Perissinitto Jr., and Orlando Lopes, On the essential spectrum of a semigroup of thermoelasticity, Nonlinear Anal. 21 (1993), no. 1, 65–75. MR 1231529, DOI https://doi.org/10.1016/0362-546X%2893%2990178-U
- Constantine M. Dafermos, On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity, Arch. Rational Mech. Anal. 29 (1968), 241–271. MR 233539, DOI https://doi.org/10.1007/BF00276727
- Daniel B. Henry, Topics in analysis, Publ. Sec. Mat. Univ. Autònoma Barcelona 31 (1987), no. 1, 29–84. MR 898603
L. Hörmander, The Analysis of Linear Partial Differential Operators 3, Springer-Verlag, New York, 1985
- Gilles Lebeau and Enrique Zuazua, Sur la décroissance non uniforme de l’énergie dans le système de la thermoélasticité linéaire, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 4, 409–415 (French, with English and French summaries). MR 1440958, DOI https://doi.org/10.1016/S0764-4442%2897%2980077-8
- S. Jiang, J. E. Muñoz Rivera, and R. Racke, Asymptotic stability and global existence in thermoelasticity with symmetry, Quart. Appl. Math. 56 (1998), no. 2, 259–275. MR 1622566, DOI https://doi.org/10.1090/qam/1622566
- Jaime E. Muñoz Rivera, Energy decay rates in linear thermoelasticity, Funkcial. Ekvac. 35 (1992), no. 1, 19–30. MR 1172418
- Franz Rellich, Darstellung der Eigenwerte von $\Delta u+\lambda u=0$ durch ein Randintegral, Math. Z. 46 (1940), 635–636 (German). MR 2456, DOI https://doi.org/10.1007/BF01181459
- Michael E. Taylor, Reflection of singularities of solutions to systems of differential equations, Comm. Pure Appl. Math. 28 (1975), no. 4, 457–478. MR 509098, DOI https://doi.org/10.1002/cpa.3160280403
M. Taylor, Partial Differential Equations II, Springer-Verlag, New York, 1996
- Enrique Zuazua, Controllability of the linear system of thermoelasticity, J. Math. Pures Appl. (9) 74 (1995), no. 4, 291–315. MR 1341768
O. Lopes, D. Henry, and A. Perissinitto, Jr., On the essential spectrum of a semigroup of thermoelasticity, Nonlinear Anal. TMA 21, No. 1, 65–75 (1993)
C. Dafermos, On the existence and asymptotic stability of solutions to the equation of linear thermoelasticity, Arch. Rational Mech. Anal. 29, 241–271 (1968)
D. Henry, Topics in analysis, Publ. Sec. Mat. Univ. Autònoma Barcelona 31, 29–84 (1987)
L. Hörmander, The Analysis of Linear Partial Differential Operators 3, Springer-Verlag, New York, 1985
G. Lebeau and E. Zuazua, Sur la décroissance non uniforme de l’énergie dans le système de la thermoélasticité linéaire, C. R. Acad. Sci., Paris, Ser. I 324, No. 4, 409–415 (1997)
S. Jiang, J. E. Muñoz Rivera, and R. Racke, Asymptotic stability and global existence in thermoelasticity with symmetry, Quart. Appl. Math. 56, 259–275 (1998)
J. E. Muñoz Rivera, Energy decay rates in linear thermoelasticity, Funkcial. Ekvac. 35, 19–30 (1992)
F. Rellich, Darstellung der Eigenwerte von $\Delta u + \lambda u$ durch ein Randintegral, Math. Z. 46, 635–636 (1940)
M. Taylor, Reflection of singularities of solutions to systems of differential equations, Comm. Pure Appl. Math. 28, 475–478 (1975)
M. Taylor, Partial Differential Equations II, Springer-Verlag, New York, 1996
E. Zuazua, Controllability of the linear system of thermoelasticity, J. Math. Pures Appl. 74, 291–315 (1995)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
74F05,
35B35,
35B40,
35Q72,
74H40
Retrieve articles in all journals
with MSC:
74F05,
35B35,
35B40,
35Q72,
74H40
Additional Information
Article copyright:
© Copyright 2000
American Mathematical Society
