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Global smooth solutions and asymptotic stability in one-dimensional nonlinear thermoelasticity. (English) Zbl 0756.73012
It is established a global existence theorem of nonlinear classical dynamic coupled thermoelasticity for a one-dimensional displacement- temperature initial-boundary value problem with homogeneous boundary conditions. The theorem asserts that for sufficiently “small” initial data there is a smooth and unique solution to the problem, and suitably defined norms of the solution vanish or are bounded as the time goes to infinity. The reviewer notes a number of slips and typographical errors, e.g.: 1. The field equations (0.3)-(0.4) are not postulated in a dimensionless form. Therefore, introducing a unit interval \(0<x<1\) as a reference configuration makes a confusion. 2. For the same reason, the definition of the norm: \(| v|_{t,K,L}\) (p. 4) involving the coefficient \((1+s)^ K\), when \(0\leq s\leq t\), and \(t\) is the time, makes another confusion. 3. The hypothesis (4.5), p. 22, in which a constant \(\sigma\) is an upper bound for the three fields of different physical dimensions, can be hardly justified.

MSC:
74A15 Thermodynamics in solid mechanics
74B20 Nonlinear elasticity
35Q72 Other PDE from mechanics (MSC2000)
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[1] Adams, R. A.: Sobolev spaces. Academic Press (1975). · Zbl 0314.46030
[2] Carlson, D. E.: Linear thermoelasticity. Handbuch der Physik VIa/2, 297-346, Springer-Verlag (1972).
[3] Chrzeszczyk, A.: Some existence results in dynamical thermoelasticity. Part I. Nonlinear case. Arch. Mech. 39 (6) (1987), 605-617. · Zbl 0682.73004
[4] Dafermos, C. M. & Hsiao, L.: Development of singularities in solutions of the equations of nonlinear thermoelasticity. Quart. Appl. Math. 44 (1986), 463-474. · Zbl 0661.35009
[5] Hrusa, W. J. & Tarabek, M. A.: On smooth solutions of the Cauchy problem in one-dimensional nonlinear thermoelasticity. Quart. Appl. Math. 47 (1989), 631-644. · Zbl 0692.73005
[6] Hrusa, W. J. & Messaoudi, S. A.: On formation of singularities in in one-dimensional nonlinear thermoelasticity. Arch. Rational Mech. Anal. 111 (1990), 135-151. · Zbl 0712.73023 · doi:10.1007/BF00375405
[7] Ikawa, M.: Mixed problems for hyperbolic equations of second order. J. Math. Soc. Japan 20 (1968), 580-608. · Zbl 0172.14304 · doi:10.2969/jmsj/02040580
[8] Jiang, S.: Global existence of smooth solutions in one-dimensional nonlinear thermoelasticity. Proc. Roy. Soc. Edinburgh 115A (1990), 257-274. · Zbl 0723.35044
[9] Jiang, S.: Global solutions of the Dirichlet problem in one-dimensional nonlinear thermoelasticity, Preprint 138, SFB 256, Univ. Bonn (1990). · Zbl 0723.35044
[10] Jiang, S. & Racke, R.: On some quasilinear hyperbolic-parabolic initial boundary value problems. Math. Meth. Appl. Sci. 12 (1990), 315-339. · Zbl 0706.35098 · doi:10.1002/mma.1670120404
[11] Kawashima, S.: Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics. Thesis, Kyoto Univ. (1983).
[12] Kawashima, S. & Okada, M.: Smooth global solutions for the one-dimensional equations in magnetohydrodynamics. Proc. Jap. Acad. 53, Ser. A (1982), 384-387. · Zbl 0522.76098
[13] Ponce, G. & Racke, R.: Global existence of solutions to the initial value problem for nonlinear thermoelasticity. J. Diff. Eqns. 87 (1990), 70-83. · Zbl 0725.35065 · doi:10.1016/0022-0396(90)90016-I
[14] Racke, R.: Blow-up in nonlinear three-dimensional thermoelasticity. Math. Meth. Appl. Sci. 12 (1990), 267-273. · Zbl 0705.35081 · doi:10.1002/mma.1670120308
[15] Racke, R.: On the Cauchy problem in nonlinear 3-d thermoelasticity. Math. Z. 203 (1990), 649-682. · Zbl 0701.73002 · doi:10.1007/BF02570763
[16] Seeley, R. T.: Integral equations depending analytically on a parameter. Indag. Math. 24 (1962), 434-442. · Zbl 0106.08102
[17] Shibata, Y.: On the global existence of classical solutions of mixed problems forsome second order non-linear hyperbolic operators with dissipative term in the interior domain. Funk. Ekva. 25 (1982), 303-345. · Zbl 0524.35070
[18] Shibata, Y.: On the global existence of classical solutions of second order fully nonlinear hyperbolic equations with first order dissipation in the exterior domain. Tsukuba J. Math. 7 (1983), 1-68. · Zbl 0524.35071
[19] Shibata, Y. & Tsutsumi, Y.: On a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain. Math. Z. 191 (1986), 165-199. · Zbl 0592.35028 · doi:10.1007/BF01164023
[20] Slemrod, M.: Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional nonlinear thermoelasticity. Arch. Rational Mech. Anal. 76 (1981), 97-133. · Zbl 0481.73009 · doi:10.1007/BF00251248
[21] Zheng, S. & Shen, W.: Global solutions to the Cauchy problem of a class of quasilinear hyperbolic parabolic coupled systems. Sci. Sinica, Ser. A, 30 (1987), 1133-1149. · Zbl 0649.35013
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