Global existence and asymptotic behaviour of the solution to the system in one-dimensional nonlinear thermoviscoelasticity

Author:
Yuming Qin

Journal:
Quart. Appl. Math. **59** (2001), 113-142

MSC:
Primary 74H20; Secondary 35B40, 35Q72, 74F05, 74H10, 74H25

DOI:
https://doi.org/10.1090/qam/1811097

MathSciNet review:
MR1811097

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper is concerned with the global existence, uniqueness, and asymptotic behavior, as time tends to infinity, of the solution to the system in nonlinear one-dimensional thermoviscoelasticity. Our results show that the global solution approaches to the solution in the norm to the corresponding stationary problem, as time tends to infinity.

**[1]**C. M. Dafermos,*Global smooth solutions to the initial boundary value problem for the equations of one-dimensional nonlinear thermoviscoelasticity*, SIAM J. Math. Anal.**13**, 397-408 (1982) MR**653464****[2]**C. M. Dafermos and L. Hsiao,*Global smooth thermomechanical processes in one-dimensional nonlinear thermoviscoelasticity*, Nonlinear Anal.**6**, 435-454 (1982) MR**661710****[3]**S. Jiang,*Global large solutions to initial boundary value problems in one-dimensional nonlinear thermoviscoelasticity*, Quart. Appl. Math.**51**, 731-744 (1992) MR**1247437****[4]**S. Jiang,*On the initial boundary value problems for a viscous, heat-conducting, one-dimensional real gas*, J. Differential Equations**110**, 157-181 (1994) MR**1278368****[5]**S. Jiang,*Global spherically symmetric solutions to the equations of a viscous polytropic ideal gas in exterior domain*, Comm. Math. Phys.**178**, 339-374 (1996) MR**1389908****[6]**S. Jiang,*Large-time behaviour of solutions to the equations of a viscous polytropic ideal gas*, Ann. Mat. Pura Appl.**175**, 253-275 (1998) MR**1748226****[7]**S. Jiang,*Large-time behaviour of solutions to the equations of a one-dimensional viscous polytropic ideal gas in unbounded domains*, Comm. Math. Phys.**200**, 181-193 (1999) MR**1671920****[8]**L. Hsiao and H. Jian,*Asymptotic behaviour of solutions to the system of one-dimensional nonlinear thermoviscoelasticity*, Chinese Ann. Math. Ser. B**19**, 143-152 (1998) MR**1655929****[9]**L. Hsiao and T. Luo,*Large-time behaviour of solutions to the equations of one-dimensional nonlinear thermoviscoelasticity*, Quart. Appl. Math.**56**, 201-219 (1998) MR**1622554****[10]**S. Kawashima and T. Nishida,*Global solutions to the initial boundary value problems for the equations of one-dimensional motion of viscous polytropic gases*, J. Math. Kyoto Univ.**21**, 825-837 (1981) MR**637519****[11]**B. Kawohl,*Global existence of large solutions to initial boundary value problems for a viscous, heat-conducting, one-dimensional real gas*, J. Differential Equations**58**, 76-103 1985) MR**791841****[12]**A. V. Kazhikhov and V. V. Shelukhin,*Unique global solution with respect to time of initial boundary value problems for one-dimensional equations of a viscous gas*, J. Appl. Math. Meth.**41**, 273-282 (1977) MR**0468593****[13]**J. U. Kim,*Global existence of solutions of the equations of one-dimensional thermoviscoelasticity with initial data in BV and*, Ann. Scuola Norm. Sup. Pisa**10**, 357-427 (1983) MR**739917****[14]**T. Nagasawa,*On the one-dimensional motion of the polytropic ideal gas non-fixed on the boundary*, J. Differential Equations**65**, 49-67 (1986) MR**859472****[15]**T. Nagasawa,*On the asymptotic behaviour of the one-dimensional motion of the polytropic ideal gas with stress-free condition*, Quart. Appl. Math.**46**, 665-679 (1988) MR**973382****[16]**M. Okada and S. Kawashima,*On the equation of one-dimensional motion of compressible viscous fluids*, J. Math. Kyoto Univ.**23**, 55-71 (1983) MR**692729****[17]**Y. Qin,*Global existence and asymptotic behaviour of solutions to a system of equations for a nonlinear one-dimensional viscous, heat-conducting real gas*, Chinese Ann. Math. Ser. A**20**(3), 343-354 (1999) MR**1712610****[18]**Y. Qin,*Global existence and asymptotic behaviour of a viscous, heat-conductive, one-dimensional real gas with fixed and thermally insulated endpoints*, Nonlinear Anal. (to appear) MR**1822093****[19]**Y. Qin,*Global existence and asymptotic behaviour for the solution to nonlinear viscous, heat-conductive, one-dimensional real gas*, Adv. Math. Sci. Appl.**10**, 119-148 (2000) MR**1769177****[20]**Y. Qin,*Global existence and asymptotic behaviour for a viscous, heat-conductive, one-dimensional real gas with fixed and constant temperature boundary conditions*, submitted to Differential and Integral Equations MR**1869558****[21]**Y. Qin,*Asymptotic behaviour for global smooth solution to a one-dimensional nonlinear thermoviscoelastic system*, J. Partial Differential Equations**12**, 111-134 (1999) MR**1705006****[22]**R. Racke and S. Zheng,*Global existence and asymptotic behaviour in nonlinear thermoviscoelasticity*, J. Differential Equations**1**, 46-67 (1997) MR**1429091****[23]**W. Shen and S. Zheng,*On the coupled Cahn-Hilliard equations*, Comm. Partial Differential Equations**18**, 701-727 (1993) MR**1214877****[24]**W. Shen and S. Zheng,*Global smooth solutions to the Cauchy problem of equations of one-dimensional nonlinear thermoviscoelasticity*, Partial Differential Equations**2**, 26-38 (1989) MR**1010771****[25]**W. Shen, S. Zheng, and P. Zhu,*Global existence and asymptotic behaviour of weak solutions to nonlinear thermoviscoelastic system with clamped boundary conditions*, Quart. Appl. Math.**57**, 93-116 (1999) MR**1672183****[26]**J. Sprekels, S. Zheng, and P. Zhu,*Asymptotic behaviour of the solutions to a Landau-Ginzburg system with viscosity for martensitic phase transitions in shape memory alloys*, SIAM J. Math. Anal.**1**, 69-84 (1998) MR**1617175****[27]**S. Zheng,*Nonlinear parabolic equations and hyperbolic-parabolic coupled systems*, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 76, Longman Group Limited, London, 1995 MR**1375458**

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
74H20,
35B40,
35Q72,
74F05,
74H10,
74H25

Retrieve articles in all journals with MSC: 74H20, 35B40, 35Q72, 74F05, 74H10, 74H25

Additional Information

DOI:
https://doi.org/10.1090/qam/1811097

Article copyright:
© Copyright 2001
American Mathematical Society