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

Qin, Yuming

doi:10.1090/qam/1811097

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.

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 (1982), no. 3, 397–408. MR 653464, DOI https://doi.org/10.1137/0513029
C. M. Dafermos and L. Hsiao, Global smooth thermomechanical processes in one-dimensional nonlinear thermoviscoelasticity, Nonlinear Anal. 6 (1982), no. 5, 435–454. MR 661710, DOI https://doi.org/10.1016/0362-546X%2882%2990058-X
Song Jiang, Global large solutions to initial-boundary value problems in one-dimensional nonlinear thermoviscoelasticity, Quart. Appl. Math. 51 (1993), no. 4, 731–744. MR 1247437, DOI https://doi.org/10.1090/qam/1247437
Song Jiang, On initial-boundary value problems for a viscous, heat-conducting, one-dimensional real gas, J. Differential Equations 110 (1994), no. 2, 157–181. MR 1278368, DOI https://doi.org/10.1006/jdeq.1994.1064
Song Jiang, Global spherically symmetric solutions to the equations of a viscous polytropic ideal gas in an exterior domain, Comm. Math. Phys. 178 (1996), no. 2, 339–374. MR 1389908
Song Jiang, Large-time behavior of solutions to the equations of a viscous polytropic ideal gas, Ann. Mat. Pura Appl. (4) 175 (1998), 253–275. MR 1748226, DOI https://doi.org/10.1007/BF01783686
Song Jiang, Large-time behavior of solutions to the equations of a one-dimensional viscous polytropic ideal gas in unbounded domains, Comm. Math. Phys. 200 (1999), no. 1, 181–193. MR 1671920, DOI https://doi.org/10.1007/s002200050526
Ling Hsiao and Huaiyu Jian, Asymptotic behavior of solutions to the system of one-dimensional nonlinear thermoviscoelasticity, Chinese Ann. Math. Ser. B 19 (1998), no. 2, 143–152. A Chinese summary appears in Chinese Ann. Math. Ser. A 19 (1998), no. 2, 283. MR 1655929
L. Hsiao and Tao Luo, Large-time behavior of solutions to the equations of one-dimensional nonlinear thermoviscoelasticity, Quart. Appl. Math. 56 (1998), no. 2, 201–219. MR 1622554, DOI https://doi.org/10.1090/qam/1622554
Shuichi Kawashima and Takaaki Nishida, Global solutions to the initial value problem for the equations of one-dimensional motion of viscous polytropic gases, J. Math. Kyoto Univ. 21 (1981), no. 4, 825–837. MR 637519, DOI https://doi.org/10.1215/kjm/1250521915
Bernhard Kawohl, Global existence of large solutions to initial-boundary value problems for a viscous, heat-conducting, one-dimensional real gas, J. Differential Equations 58 (1985), no. 1, 76–103. MR 791841, DOI https://doi.org/10.1016/0022-0396%2885%2990023-3
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, Prikl. Mat. Meh. 41 (1977), no. 2, 282–291 (Russian); English transl., J. Appl. Math. Mech. 41 (1977), no. 2, 273–282. MR 0468593, DOI https://doi.org/10.1016/0021-8928%2877%2990011-9
Jong Uhn Kim, Global existence of solutions of the equations of one-dimensional thermoviscoelasticity with initial data in ${\rm BV}$ and $L^{1}$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 10 (1983), no. 3, 357–427. MR 739917
Takeyuki Nagasawa, On the one-dimensional motion of the polytropic ideal gas nonfixed on the boundary, J. Differential Equations 65 (1986), no. 1, 49–67. MR 859472, DOI https://doi.org/10.1016/0022-0396%2886%2990041-0
Takeyuki Nagasawa, On the asymptotic behavior of the one-dimensional motion of the polytropic ideal gas with stress-free condition, Quart. Appl. Math. 46 (1988), no. 4, 665–679. MR 973382, DOI https://doi.org/10.1090/S0033-569X-1988-0973382-3
Mari Okada and Shuichi Kawashima, On the equations of one-dimensional motion of compressible viscous fluids, J. Math. Kyoto Univ. 23 (1983), no. 1, 55–71. MR 692729, DOI https://doi.org/10.1215/kjm/1250521610
Yu Ming Qin, Global existence and asymptotic behavior of solutions to a system of equations for a nonlinear one-dimensional viscous heat-conducting real gas, Chinese Ann. Math. Ser. A 20 (1999), no. 3, 343–354 (Chinese, with Chinese summary). MR 1712610
Yuming Qin, Global existence and asymptotic behavior for a viscous, heat-conductive, one-dimensional real gas with fixed and thermally insulated endpoints, Nonlinear Anal. 44 (2001), no. 4, Ser. A: Theory Methods, 413–441. MR 1822093, DOI https://doi.org/10.1016/S0362-546X%2899%2900140-6
Yuming Qin, Global existence and asymptotic behaviour for the solution to nonlinear viscous, heat-conductive, one-dimensional real gas, Adv. Math. Sci. Appl. 10 (2000), no. 1, 119–148. MR 1769177
Yuming Qin, Global existence and asymptotic behavior for a viscous, heat-conductive, one-dimensional real gas with fixed and constant temperature boundary conditions, Adv. Differential Equations 7 (2002), no. 2, 129–154. MR 1869558
Yuming Qin, Asymptotic behavior for global smooth solution to a one-dimensional nonlinear thermoviscoelastic system, J. Partial Differential Equations 12 (1999), no. 2, 111–134. MR 1705006
Reinhard Racke and Songmu Zheng, Global existence and asymptotic behavior in nonlinear thermoviscoelasticity, J. Differential Equations 134 (1997), no. 1, 46–67. MR 1429091, DOI https://doi.org/10.1006/jdeq.1996.3216
Wei Xi Shen and Song Mu Zheng, On the coupled Cahn-Hilliard equations, Comm. Partial Differential Equations 18 (1993), no. 3-4, 701–727. MR 1214877, DOI https://doi.org/10.1080/03605309308820946
Song Mu Zheng and Wei Xi Shen, Global smooth solutions to Cauchy problem of equations of one-dimensional thermoviscoelasticity, J. Partial Differential Equations 2 (1989), no. 2, 26–38. MR 1010771
Weixi Shen, Songmu Zheng, and Peicheng Zhu, Global existence and asymptotic behavior of weak solutions to nonlinear thermoviscoelastic systems with clamped boundary conditions, Quart. Appl. Math. 57 (1999), no. 1, 93–116. MR 1672183, DOI https://doi.org/10.1090/qam/1672183
Jürgen Sprekels, Songmu Zheng, and Peicheng Zhu, Asymptotic behavior of the solutions to a Landau-Ginzburg system with viscosity for martensitic phase transitions in shape memory alloys, SIAM J. Math. Anal. 29 (1998), no. 1, 69–84. MR 1617175, DOI https://doi.org/10.1137/S0036141096297698
Songmu Zheng, Nonlinear parabolic equations and hyperbolic-parabolic coupled systems, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 76, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1995. MR 1375458