Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

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 $ {H^{1}}$ norm to the corresponding stationary problem, as time tends to infinity.


References [Enhancements On Off] (What's this?)

  • [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 $ L^{l}$, 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

Similar Articles

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

American Mathematical Society