Global solvability and exponential stability in one-dimensional nonlinear thermoelasticity
Authors:
Reinhard Racke, Yoshihiro Shibata and Song Mu Zheng
Journal:
Quart. Appl. Math. 51 (1993), 751-763
MSC:
Primary 35Q72; Secondary 35B35, 73B30, 73C50
DOI:
https://doi.org/10.1090/qam/1247439
MathSciNet review:
MR1247439
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We are mainly concerned with the Dirichlet initial boundary value problem in one-dimensional nonlinear thermoelasticity. It is proved that if the initial data are close to the equilibrium then the problem admits a unique, global, smooth solution. Moreover, as time tends to infinity, the solution is exponentially stable. As a corollary we also obtain the existence of periodic solutions for small, periodic righthand sides.
- 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 J. Hrusa and Salim A. Messaoudi, On formation of singularities in one-dimensional nonlinear thermoelasticity, Arch. Rational Mech. Anal. 111 (1990), no. 2, 135–151. MR 1057652, DOI https://doi.org/10.1007/BF00375405
- William J. Hrusa and Michael A. Tarabek, On smooth solutions of the Cauchy problem in one-dimensional nonlinear thermoelasticity, Quart. Appl. Math. 47 (1989), no. 4, 631–644. MR 1031681, DOI https://doi.org/10.1090/qam/1031681
- Song Jiang, Global existence of smooth solutions in one-dimensional nonlinear thermoelasticity, Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), no. 3-4, 257–274. MR 1069521, DOI https://doi.org/10.1017/S0308210500020631
- Song Jiang, Global solutions of the Neumann problem in one-dimensional nonlinear thermoelasticity, Nonlinear Anal. 19 (1992), no. 2, 107–121. MR 1174462, DOI https://doi.org/10.1016/0362-546X%2892%2990114-T
- Song Jiang, Global solutions of the Neumann problem in one-dimensional nonlinear thermoelasticity, Nonlinear Anal. 19 (1992), no. 2, 107–121. MR 1174462, DOI https://doi.org/10.1016/0362-546X%2892%2990114-T
S. Kawashima, Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics, Thesis, Kyoto University, 1983
- Shuichi Kawashima and Mari Okada, Smooth global solutions for the one-dimensional equations in magnetohydrodynamics, Proc. Japan Acad. Ser. A Math. Sci. 58 (1982), no. 9, 384–387. MR 694940
- S. Klainerman and Gustavo Ponce, Global, small amplitude solutions to nonlinear evolution equations, Comm. Pure Appl. Math. 36 (1983), no. 1, 133–141. MR 680085, DOI https://doi.org/10.1002/cpa.3160360106
- 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
- Jaime E. Muñoz Rivera, Energy decay rates in linear thermoelasticity, Funkcial. Ekvac. 35 (1992), no. 1, 19–30. MR 1172418
- Reinhard Racke and Yoshihiro Shibata, Global smooth solutions and asymptotic stability in one-dimensional nonlinear thermoelasticity, Arch. Rational Mech. Anal. 116 (1991), no. 1, 1–34. MR 1130241, DOI https://doi.org/10.1007/BF00375601
- Yoshihiro Shibata, Neumann problem for one-dimensional nonlinear thermoelasticity, Partial differential equations, Part 1, 2 (Warsaw, 1990) Banach Center Publ., 27, Part 1, vol. 2, Polish Acad. Sci. Inst. Math., Warsaw, 1992, pp. 457–480. MR 1205848
- Yoshihiro Shibata and Yoshio Tsutsumi, On a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain, Math. Z. 191 (1986), no. 2, 165–199. MR 818663, DOI https://doi.org/10.1007/BF01164023
- 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
- Song Mu Zheng, Global solutions and applications to a class of quasilinear hyperbolic-parabolic coupled systems, Sci. Sinica Ser. A 27 (1984), no. 12, 1274–1286. MR 794293
- Song Mu Zheng and Wei Xi Shen, Global solutions to the Cauchy problem of quasilinear hyperbolic parabolic coupled systems, Sci. Sinica Ser. A 30 (1987), no. 11, 1133–1149. MR 942420
C. M. Dafermos and L. Hsiao, Development of singularities in solutions of the equations of nonlinear thermoelasticity, Quart. Appl. Math. 44, 463–474 (1986)
W. H. Hrusa and S. A. Messaoudi, On formation of singularities in one-dimensional nonlinear thermoelasticity, Arch. Rational Mech. Anal. 111, 135–151 (1990)
W. J. Hrusa and M. A. Tarabek, On smooth solutions of the Cauchy problem in one-dimensional nonlinear thermoelasticity, Quart. Appl. Math. 47, 631–644 (1989)
S. Jiang, Global existence of smooth solutions in one-dimensional nonlinear thermoelasticity, Proc. Roy. Soc. Edinburgh Sect. A 115, 257–274 (1990)
S. Jiang, Global solutions of the Dirichlet problem in one-dimensional nonlinear thermoelasticity, SFB 256 preprint 138, Universität Bonn, 1990
S. Jiang, Global solutions of the Neumann problem in one-dimensional nonlinear thermoelasticity, Nonlinear Anal. 19, 107–121 (1992)
S. Kawashima, Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics, Thesis, Kyoto University, 1983
S. Kawashima and M. Okada, Smooth global solutions for the one-dimensional equations in magnetohydrodynamics, Proc. Japan Acad. Ser. A Math. Sci. 53, 384–387 (1982)
S. Klainerman and G. Ponce, Global, small amplitude solutions to nonlinear evolution equations, Comm. Pure Appl. Math. 36, 133–141 (1983)
A. 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, 349–379 (1977)
J. E. Muñoz Rivera, Energy decay rates in linear thermoelasticity, Funkcial. Ekvac 35, 19–30 (1992)
R. Racke and Y. Shibata, Global smooth solutions and asymptotic stability in one-dimensional nonlinear thermoelasticity, Arch. Rational Mech. Anal. 116, 1–34 (1991)
Y. Shibata, Neumann problem for one-dimensional nonlinear thermoelasticity, SFB 256 preprint 145, Universität Bonn, 1990
Y. Shibata and Y. Tsutsumi, On a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain, Math. Z. 191, 165–199 (1986)
M. Slemrod, Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional non-linear thermoelasticity, Arch. Rational Mech. Anal. 76, 97–133 (1981)
S. Zheng, Global solutions and applications to a class of quasilinear hyperbolic-parabolic coupled systems, Sci. Sinica Ser. A 127, 1274–1286 (1984)
S. Zheng and W. Shen, Global solutions to the Cauchy problem of quasilinear hyperbolic parabolic coupled systems, Sci. Sinica Ser. A 30, 1133–1149 (1987)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
35Q72,
35B35,
73B30,
73C50
Retrieve articles in all journals
with MSC:
35Q72,
35B35,
73B30,
73C50
Additional Information
Article copyright:
© Copyright 1993
American Mathematical Society