𝐿^{𝑝}-𝐿^{𝑞} decay estimates for the Cauchy problem of linear thermoelastic systems with second sound in one space variable

Yang, Lin; Wang, Ya-Guang

doi:10.1090/S0033-569X-06-00989-6

$L^{p}-L^{q}$ decay estimates for the Cauchy problem of linear thermoelastic systems with second sound in one space variable

Authors: Lin Yang and Ya-Guang Wang
Journal: Quart. Appl. Math. 64 (2006), 1-15
MSC (2000): Primary 35Q30, 35A21
DOI: https://doi.org/10.1090/S0033-569X-06-00989-6
Published electronically: January 24, 2006
MathSciNet review: 2211374
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Abstract: $L^p-L^q$ decay estimates of solutions to the Cauchy problem of linear thermoelastic systems with second sound in one space variable will be studied in this paper. First, by dividing the frequency of phase space of the Fourier transformation into different regions, the asymptotic behavior of characteristic roots of the coefficient matrix is obtained by carefully analyzing the effect of the different regions. Second, with the help of the information on the characteristic roots and by using the interpolation theorem, the $L^{p}-L^{q}$ decay estimate of solutions to the Cauchy problem of the linear thermoelastic system with second sound in one space variable is obtained.

Carlo Cattaneo, Sulla conduzione del calore, Atti Sem. Mat. Fis. Univ. Modena 3 (1949), 83–101 (Italian). MR 0032898

2 D. S. Chandrasekharaiah: Thermoelasticity with second sound: a review. Appl. Mech. Rev. 39(1986), 355-376.

3 D. S. Chandrasekharaiah: 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

5 M. E. Gurtin and A. C. Pipkin: A general theory of heat conduction with finite wave speeds. Arch. Rat. Mech. Anal., 31(1968), 113-126.

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
D. D. Joseph and Luigi Preziosi, Heat waves, Rev. Modern Phys. 61 (1989), no. 1, 41–73. MR 977943, DOI https://doi.org/10.1103/RevModPhys.61.41

8 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, Lectures on nonlinear evolution equations, Aspects of Mathematics, E19, Friedr. Vieweg & Sohn, Braunschweig, 1992. Initial value problems. MR 1158463
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
Michael Reissig and Ya-Guang Wang, Cauchy problems for linear thermoelastic systems of type III in one space variable, Math. Methods Appl. Sci. 28 (2005), no. 11, 1359–1381. MR 2150160, DOI https://doi.org/10.1002/mma.619
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

15 X. Wang and X. Xu: Thermoelastic wave induced by pulsed laser heating. Appl. Phys., A, 73(2001), 107-114.

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 and Michael Reissig, Parabolic type decay rates for 1-D-thermoelastic systems with time-dependent coefficients, Monatsh. Math. 138 (2003), no. 3, 239–259. MR 1969519, DOI https://doi.org/10.1007/s00605-002-0506-z
Lin Yang and Ya-Guang Wang, Propagation of singularities in Cauchy problems for quasilinear thermoelastic systems in three space variables, J. Math. Anal. Appl. 291 (2004), no. 2, 638–652. MR 2039075, DOI https://doi.org/10.1016/j.jmaa.2003.11.032
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