Asymptotic behavior of solutions in linear 2- or 3-D thermoelasticity with second sound
Author:
Reinhard Racke
Journal:
Quart. Appl. Math. 61 (2003), 315-328
MSC:
Primary 74H40; Secondary 35B35, 35Q72, 74H05
DOI:
https://doi.org/10.1090/qam/1976372
MathSciNet review:
MR1976372
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Abstract: We consider thermoelastic systems in two or three space dimensions where thermal disturbances are modeled propagating as wavelike pulses traveling at finite speed. This is done using Cattaneo’s law for heat conduction instead of Fourier’s law. For Dirichlet type boundary conditions, the exponential stability of the now purely, but slightly damped, hyperbolic system is proved in the radially symmetric case.
D. E. Carlson, Linear thermoelasticity, Handbuch der Physik VIa/2 (1972), 297–346
D. S. Chandrasekharaiah, Hyperbolic thermoelasticity: a review of recent literature, Appl. Mech. Rev. 51 (1998), 705–729
- 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
- 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
- Song Jiang and Reinhard Racke, Evolution equations in thermoelasticity, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 112, Chapman & Hall/CRC, Boca Raton, FL, 2000. MR 1774100
- Herbert Koch, Slow decay in linear thermoelasticity, Quart. Appl. Math. 58 (2000), no. 4, 601–612. MR 1788420, DOI https://doi.org/10.1090/qam/1788420
- Gilles Lebeau and Enrique Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity, Arch. Ration. Mech. Anal. 148 (1999), no. 3, 179–231. MR 1716306, DOI https://doi.org/10.1007/s002050050160
- Rolf Leis, Initial-boundary value problems in mathematical physics, B. G. Teubner, Stuttgart; John Wiley & Sons, Ltd., Chichester, 1986. MR 841971
- Zhuangyi Liu and Songmu Zheng, Semigroups associated with dissipative systems, Chapman & Hall/CRC Research Notes in Mathematics, vol. 398, Chapman & Hall/CRC, Boca Raton, FL, 1999. MR 1681343
H. W. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids 15 (1967), 299–309
M. Mosbacher, V. Dobler, H.-.J. Münzer, J. Zimmermann, J. Solis, J. Boneberg, and P. Leiderer, Optical field enhancement effects in laser assisted particle removal, Appl. Phys. A 72 (1991), 41–44
- Reinhard Racke, On the time-asymptotic behaviour of solutions in thermoelasticity, Proc. Roy. Soc. Edinburgh Sect. A 107 (1987), no. 3-4, 289–298. MR 924522, DOI https://doi.org/10.1017/S0308210500031164
R. Racke, Thermoelasticity with second sound—Exponential stability in linear and nonlinear 1-d, Konstanzer Schriften Math. Inf. 141 (2001)
- T. Sabri Öncü and T. Bryant Moodie, On the constitutive relations for second sound in elastic solids, Arch. Rational Mech. Anal. 121 (1992), no. 1, 87–99. MR 1185571, DOI https://doi.org/10.1007/BF00375440
- Hany H. Sherief, On uniqueness and stability in generalized thermoelasticity, Quart. Appl. Math. 44 (1987), no. 4, 773–778. MR 872828, DOI https://doi.org/10.1090/S0033-569X-1987-0872828-9
K. K. Tamma and R. R. Namburu, Computational approaches with applications to nonclassical and classical thermomechanical problems, Appl. Mech. Rev. 50 (1997), 514–551
- Michael A. Tarabek, On the existence of smooth solutions in one-dimensional nonlinear thermoelasticity with second sound, Quart. Appl. Math. 50 (1992), no. 4, 727–742. MR 1193663, DOI https://doi.org/10.1090/qam/1193663
X. Wang and X. Xu, Thermoelastic wave induced by pulsed laser heating, Appl. Phys. A (to appear). DOI: 10.1007/s003390000593
- Songmu Zheng, Nonlinear parabolic equations and hyperbolic-parabolic coupled systems, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 76, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1995. MR 1375458
D. E. Carlson, Linear thermoelasticity, Handbuch der Physik VIa/2 (1972), 297–346
D. S. Chandrasekharaiah, Hyperbolic thermoelasticity: a review of recent literature, Appl. Mech. Rev. 51 (1998), 705–729
C. 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
S. Jiang, J. E. Muñoz Rivera, and R. Racke, Asymptotic stability and global existence in thermoelasticity with symmetry, Quart. Appl. Math. 56 (1998), 259–275
S. Jiang and R. Racke, Evolution equations in thermoelasticity, $\pi$ Monographs Surveys Pure Appl. Math. 112, Chapman & Hall/CRC, Boca Raton (2000)
H. Koch, Slow decay in linear thermoelasticity, Quart. Appl. Math. 58 (2000), 601–612
G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity, Arch. Rational Mech. Anal. 148 (1999), 179–231
R. Leis, Initial boundary value problems in mathematical physics, B. G. Teubner-Verlag, Stuttgart; John Wiley & Sons, Chichester (1986)
Z. Liu and S. Zheng, Semigroups associated with dissipative systems, Chapman & Hall/CRC Res. Notes Math. 398 (1999)
H. W. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids 15 (1967), 299–309
M. Mosbacher, V. Dobler, H.-.J. Münzer, J. Zimmermann, J. Solis, J. Boneberg, and P. Leiderer, Optical field enhancement effects in laser assisted particle removal, Appl. Phys. A 72 (1991), 41–44
R. Racke, On the time-asymptotic behaviour of solutions in thermoelasticity, Proc. Roy. Soc. Edinburgh 107A (1987), 289–298.
R. Racke, Thermoelasticity with second sound—Exponential stability in linear and nonlinear 1-d, Konstanzer Schriften Math. Inf. 141 (2001)
T. Sabir Öncü and T. Bryant Moodie, On the constitutive relations for second sound in elastic solids, Arch. Rational Mech. Anal. 121 (1992), 87–99
H. Sherief, On uniqueness and stability in generalized thermoelasticity, Quart. Appl. Math. 44 (1987), 773–778
K. K. Tamma and R. R. Namburu, Computational approaches with applications to nonclassical and classical thermomechanical problems, Appl. Mech. Rev. 50 (1997), 514–551
M. A. Tarabek, On the existence of smooth solutions in one-dimensional nonlinear thermoelasticity with second sound, Quart. Appl. Math. 50 (1992), 727–742
X. Wang and X. Xu, Thermoelastic wave induced by pulsed laser heating, Appl. Phys. A (to appear). DOI: 10.1007/s003390000593
S. Zheng, Nonlinear parabolic equations and hyperbolic-parabolic coupled systems, Pitman Monographs Surv. Pure Appl. Math. 76, Longman; John Wiley & Sons, New York (1995)
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© Copyright 2003
American Mathematical Society
