On smooth solutions of the Cauchy problem in one-dimensional nonlinear thermoelasticity
Authors:
William J. Hrusa and Michael A. Tarabek
Journal:
Quart. Appl. Math. 47 (1989), 631-644
MSC:
Primary 35Q20; Secondary 73B30, 73U05
DOI:
https://doi.org/10.1090/qam/1031681
MathSciNet review:
MR1031681
Full-text PDF Free Access
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Additional Information
- Bernard D. Coleman and Morton E. Gurtin, Waves in materials with memory. III. Thermodymanic influences on the growth and decay of acceleration waves, Arch. Rational Mech. Anal. 19 (1965), 266–298. MR 195337, DOI https://doi.org/10.1007/BF00250214
- Bernard D. Coleman and Walter Noll, The thermodynamics of elastic materials with heat conduction and viscosity, Arch. Rational Mech. Anal. 13 (1963), 167–178. MR 153153, DOI https://doi.org/10.1007/BF01262690
- Bernard D. Coleman and Victor J. Mizel, Thermodynamics and departures from Fourier’s law of heat conduction, Arch. Rational Mech. Anal. 13 (1963), 245–261. MR 151119, DOI https://doi.org/10.1007/BF01262695
- C. M. Dafermos and L. Hsiao, Development of singularities in solutions of the equations of nonlinear thermoelasticity, Quart. Appl. Math. 44 (1986), no. 3, 463–474. MR 860899, DOI https://doi.org/10.1090/S0033-569X-1986-0860899-8
- William Alan Day, A commentary on thermodynamics, Springer Tracts in Natural Philosophy, vol. 32, Springer-Verlag, New York, 1988. MR 918561
- Peter D. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Mathematical Phys. 5 (1964), 611–613. MR 165243, DOI https://doi.org/10.1063/1.1704154
- R. C. MacCamy and V. J. Mizel, Existence and nonexistence in the large of solutions of quasilinear wave equations, Arch. Rational Mech. Anal. 25 (1967), 299–320. MR 216165, DOI https://doi.org/10.1007/BF00250932
- Akitaka Matsumura, Global existence and asymptotics of the solutions of the second-order quasilinear hyperbolic equations with the first-order dissipation, Publ. Res. Inst. Math. Sci. 13 (1977/78), no. 2, 349–379. MR 0470507, DOI https://doi.org/10.2977/prims/1195189813
- M. Slemrod, Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional nonlinear thermoelasticity, Arch. Rational Mech. Anal. 76 (1981), no. 2, 97–133. MR 629700, DOI https://doi.org/10.1007/BF00251248
S. Zheng and W. Shen, L$^{P}$ decay estimates of solutions to the Cauchy problem of hyperbolic-parabolic coupled systems, Scientia Sinica (to appear)
- Song Mu Zheng and Wei Xi Shen, Global solutions to the Cauchy problem of a class of quasilinear hyperbolic parabolic coupled systems, International workshop on applied differential equations (Beijing, 1985) World Sci. Publishing, Singapore, 1986, pp. 335–338. MR 901344
- Song Mu Zheng and Wei Xi Shen, Global solutions to the Cauchy problem of a class of quasilinear hyperbolic parabolic coupled systems, International workshop on applied differential equations (Beijing, 1985) World Sci. Publishing, Singapore, 1986, pp. 335–338. MR 901344
B. D. Coleman and M. E. Gurtin, Waves in materials with memory, III. Thermodynamic influences on the growth and decay of acceleration waves, Arch. Rational Mech. Anal. 19, 266–298 (1965)
B. D. Coleman and W. Noll, The thermodynamics of elastic materials with heat condition and viscosity, Arch. Rational Mech. Anal. 13, 167–178 (1963)
B. D. Coleman and V. J. Mizel, Thermodynamics and departure from Fourier’s law of heat conduction, Arch. Rational Mech. Anal. 13, 245–261 (1963)
C. M. Dafermos and L. Hsiao, Development of singularities in solutions of equations of nonlinear thermoelasticity, Quart. Appl. Math. 44, 463–474 (1986)
W. A. Day, A commentary on Thermodynamics, Springer-Verlag, 1988
P. D. Lax, Development of singularities of solutions of nonlinear hyperbolic differential equations, J. Math. Physics 5, 611–613 (1964)
R. C. MacCamy and V. J. Mizel, Existence and non-existence in the large of solutions of quasilinear wave equations, Arch. Rational. Mech. Anal. 25, 299–320 (1967)
A. Matsumura, Global existence and asymptotics of the solutions of the second-order quasilinear hyperbolic equations with first order dissipation, Publ. Res. Inst. Math. Sci., Kyoto University, Series A 13, 349–379 (1977)
M. Slemrod, Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional nonlinear thermoelasticity, Arch. Rational Mech. Anal. 76, 97–133 (1981)
S. Zheng and W. Shen, L$^{P}$ decay estimates of solutions to the Cauchy problem of hyperbolic-parabolic coupled systems, Scientia Sinica (to appear)
S. Zheng and W. Shen, Global solutions to the Cauchy problem of a class of hyperbolic-parabolic coupled systems, Scientia Sinica (to appear)
S. Zheng and W. Shen, Global solutions to the Cauchy problem of a class of hyperbolic-parabolic coupled systems, International Workshop on Applied Differential Equations (S. T. Xiao and F. Q. Pu, eds.). World Scientific Publishing, 1986, pp. 335–338
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Article copyright:
© Copyright 1989
American Mathematical Society