Asymptotic behavior of discontinuous solutions in 3-D thermoelasticity with second sound
Authors:
Reinhard Racke and Ya-Guang Wang
Journal:
Quart. Appl. Math. 66 (2008), 707-724
MSC (2000):
Primary 35B40, 74F05
DOI:
https://doi.org/10.1090/S0033-569X-08-01121-2
Published electronically:
October 7, 2008
MathSciNet review:
2465141
Full-text PDF Free Access
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Abstract: This paper is devoted to the study of the Cauchy problem for linear and semilinear thermoelastic systems with second sound in three space dimensions with discontinuous initial data. Due to Cattaneo’s law, replacing Fourier’s law for heat conduction, the thermoelastic system with second sound is hyperbolic. We investigate the behavior of discontinuous solutions as the relaxation parameter tends to zero, which corresponds to a formal convergence of the system to the hyperbolic-parabolic type of classical thermoelasticity. By studying expansions with respect to the relaxation parameter of the jumps of the potential part of the system on the evolving characteristic surfaces, we obtain that the jump of the temperature goes to zero while the jumps of the heat flux and the gradient of the potential part of the elastic wave are propagated along the characteristic curves of the elastic fields when the relaxation parameter goes to zero. An interesting phenomenon is when time goes to infinity: the behavior will depend on the mean curvature of the initial surface of discontinuity. These jumps decay exponentially when time goes to infinity, more rapidly for a smaller heat conductive coefficient in linear problems and in nonlinear problems when certain growth conditions are imposed on the nonlinear functions.
References
- Michael Beals, Propagation and interaction of singularities in nonlinear hyperbolic problems, Progress in Nonlinear Differential Equations and their Applications, vol. 3, Birkhäuser Boston, Inc., Boston, MA, 1989. MR 1033737
- Chandrasekharaiah, D. S.: Hyperbolic thermoelasticity: A review of recent literature. Appl. Mech. Rev. 51 (1998), 705–729.
- Shuxing Chen and Ya-Guang Wang, Propagation of singularities of solutions to hyperbolic-parabolic coupled systems, Math. Nachr. 242 (2002), 46–60. MR 1916849, DOI https://doi.org/10.1002/1522-2616%28200207%29242%3A1%3C46%3A%3AAID-MANA46%3E3.0.CO%3B2-F
- David Hoff, Dynamics of singularity surfaces for compressible, viscous flows in two space dimensions, Comm. Pure Appl. Math. 55 (2002), no. 11, 1365–1407. MR 1916987, DOI https://doi.org/10.1002/cpa.10046
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190
- 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
- Guy Métivier, Propagation, interaction and reflection of discontinuous progressing waves for semilinear hyperbolic systems, Amer. J. Math. 111 (1989), no. 2, 239–287. MR 987758, DOI https://doi.org/10.2307/2374510
- Reinhard Racke, Thermoelasticity with second sound—exponential stability in linear and non-linear 1-d, Math. Methods Appl. Sci. 25 (2002), no. 5, 409–441. MR 1888164, DOI https://doi.org/10.1002/mma.298
- Reinhard Racke, Asymptotic behavior of solutions in linear 2- or 3-D thermoelasticity with second sound, Quart. Appl. Math. 61 (2003), no. 2, 315–328. MR 1976372, DOI https://doi.org/10.1090/qam/1976372
- Reinhard Racke and Ya-Guang Wang, Propagation of singularities in one-dimensional thermoelasticity, J. Math. Anal. Appl. 223 (1998), no. 1, 216–247. MR 1627324, DOI https://doi.org/10.1006/jmaa.1998.5972
- Reinhard Racke and Ya-Guang Wang, Asymptotic behavior of discontinuous solutions to thermoelastic systems with second sound, Z. Anal. Anwendungen 24 (2005), no. 1, 117–135. MR 2146553, DOI https://doi.org/10.4171/ZAA/1232
- Jeffrey Rauch and Michael C. Reed, Discontinuous progressing waves for semilinear systems, Comm. Partial Differential Equations 10 (1985), no. 9, 1033–1075. MR 806255, DOI https://doi.org/10.1080/03605308508820400
- Ya-Guang Wang, Microlocal analysis in nonlinear thermoelasticity, Nonlinear Anal. 54 (2003), no. 4, 683–705. MR 1983443, DOI https://doi.org/10.1016/S0362-546X%2803%2900095-6
- Ya-Guang Wang, A new approach to study hyperbolic-parabolic coupled systems, Evolution equations (Warsaw, 2001) Banach Center Publ., vol. 60, Polish Acad. Sci. Inst. Math., Warsaw, 2003, pp. 227–236. MR 1993072
References
- Beals, M.: Propagation and Interaction of Singularities in Nonlinear Hyperbolic Problems. Birkhäuser, Boston, 1989. MR 1033737 (91a:58183)
- Chandrasekharaiah, D. S.: Hyperbolic thermoelasticity: A review of recent literature. Appl. Mech. Rev. 51 (1998), 705–729.
- Chen, S. and Wang, Y. G.: Propagation of singularities of solutions to hyperbolic-parabolic coupled systems. Math. Nachr., 242(2002), 46-60. MR 1916849 (2003e:35008)
- Hoff, D.: Dynamics of singularity surfaces for compressible, viscous flows in two space dimensions. Commun. Pure Appl. Math. 55 (2002), 1365–1407. MR 1916987 (2003k:76108)
- Gilbarg, D. and Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, Grundlehren math. Wiss. 224, Springer-Verlag, Berlin (1983). MR 737190 (86c:35035)
- Jiang, S. and Racke, R.: Evolution Equations in Thermoelasticity, Monographs and Surveys in Pure Appl. Math. 112, Chapman & Hall/CRC, Boca Raton (2000). MR 1774100 (2001g:74013)
- Métivier, G.: Propagation, interaction and reflection of discontinuous progressing waves for semilinear hyperbolic systems. Amer. J. Math. 111 (1989), 239–287. MR 987758 (90g:35097)
- Racke, R.: Thermoelasticity with second sound — exponential stability in linear and non-linear 1-d. Math. Meth. Appl. Sci. 25 (2002), 409–441. MR 1888164 (2002m:74016)
- Racke, R.: Asymptotic behavior of solutions in linear 2- or 3-d thermoelasticity with second sound. Quart. Appl. Math. 61 (2003), 315–328. MR 1976372 (2004a:74026)
- Racke, R. and Wang, Y. G.: Propagation of singularities in one-dimensional thermoelasticity. J. Math. Anal. Appl. 223 (1998), 216–247. MR 1627324 (99f:73014)
- Racke, R. and Wang, Y. G.: Asymptotic behavior of discontinuous solutions to thermoelastic systems with second sound. J. Anal. Appl. 24 (2005), 117–135. MR 2146553 (2006k:74040)
- Rauch, J. and Reed, M.: Discontinuous progressing waves for semilinear systems. Commun. PDE 10 (1985), 1033–1075. MR 806255 (87g:35146)
- Wang, Y.G.: Microlocal analysis in nonlinear thermoelasticity. Nonlinear Anal. 54 (2003), 683–705. MR 1983443 (2004b:74022)
- Wang, Y.G.: A new approach to study hyperbolic-parabolic coupled systems. In “Evolution Equations” (R. Picard, M. Reissig & W. Zajaczkowski, eds.), Banach Center Publications, 60(2003), 227-236. MR 1993072 (2004h:35164)
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Additional Information
Reinhard Racke
Affiliation:
Department of Mathematics and Statistics, University of Konstanz, 78457 Konstanz, Germany
Email:
reinhard.racke@uni-konstanz.de
Ya-Guang Wang
Affiliation:
Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
MR Author ID:
291072
Email:
ygwang@sjtu.edu.cn
Keywords:
Hyperbolic thermoelasticity,
semilinear,
discontinuous solutions,
asymptotic behavior,
curvature
Received by editor(s):
May 20, 2007
Published electronically:
October 7, 2008
Article copyright:
© Copyright 2008
Brown University
The copyright for this article reverts to public domain 28 years after publication.