Asymptotic behavior of the solutions to a Landau-Ginzburg system with viscosity for martensitic phase transitions in shape memory alloys

SIAM Journal on Mathematical Analysis

Volumn 29, Issue 1, 1998, Pages 69-84

  Sprekels, Jürgen ^a   Zheng, Songmu ^b   Zhu, Peicheng ^b  

^a WEIERSTRASS INSTITUTE FOR APPLIED ANALYSIS AND STOCHASTICS   (Germany)

^b FUDAN UNIVERSITY   (China)

Author keywords

Asymptotic behavior; Compact orbits; Landau Ginzburg theory; Nonlinear thermoviscoelasticity; Phase transitions; Shape memory alloys

Indexed keywords

EID: 0032343438     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036141096297698     Document Type: Article

References (22)

1. M. ACHENBACH AND I. MÜLLER, Creep and yield in martensitic transformations, Ingenieur-Archiv, 53 (1983), pp. 73-83.
2. x, J. Differential Equations, 35 (1980), pp. 200-231.
3. G. ANDREWS AND J. M. BALL, Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity, J. Differential Equations, 44 (1982), pp. 306-341.
4. M. BROKATE AND J. SPREKELS, Hysteresis and Phase Transitions, Appl. Math. Sci. 121, Springer, New York, 1996.
5. 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), pp. 397-408.
6. C. M. DAFERMOS AND L. HSIAO, Global smooth thermomechanical processes in one-dimensional nonlinear thermoviscoelasticity, Nonlinear Anal., 6 (1982), pp. 435-454.
7. F. FALK, Ginzburg-Landau theory of static domain walls in shape memory alloys, Physica B, 51 (1983), pp. 177-185.
8. F. FALK, Ginzburg-Landau theory and solitary waves in shape memory alloys, Physica B, 54 (1984), pp. 159-167.
9. Z. CHEN AND K.-H. HOFFMANN, On a one-dimensional nonlinear thermoviscoelastic model for structural phase transitions in shape memory alloys, J. Differential Equations, 112 (1994), pp. 325-350.
10. K.-H. HOFFMANN AND A. ZOCHOWSKI, Existence of solutions to some non-linear thermoelastic systems with viscosity, Math. Meth. Appl. Sci., 15 (1992), pp. 187-204.
11. S. JIANG, Global large solutions to initial boundary value problems in one-dimensional nonlinear thermoviscoelasticity, Quart. Appl. Math., 51 (1993), pp. 731-744.
12. S. LUCKHAUS AND S. ZHENG, A nonlinear boundary value problem involving a nonlocal term, Nonlinear Anal., 22 (1994), pp. 129-135.
13. T. LUO, Qualitative Behavior to Nonlinear Evolution Equations with Dissipation, Ph.D. thesis, Institute of Mathematics, Academy of Sciences of China, Beijing, 1994.
14. I. MÜLLER AND K. WILMAŃSKI, A model for phase transitions in pseudoelastic bodies, Nuovo Cimento B, 57 (1980), pp. 283-318.
15. M. NIEZGÓDKA AND J. SPREKELS, Existence of solutions for a mathematical model of structural phase transitions in shape memory alloys, Math. Meth. Appl. Sci., 10 (1988), pp. 197-223.
16. A. NOVICK-COHEN AND S. ZHENG, The Penrose-Fife type equations: Counting the one-dimensional stationary solutions, Proc. Royal Soc. Edinburgh Ser. A, 126 (1996), pp. 483-504.
17. R. L. PEGO, Phase transitions in one-dimensional nonlinear viscoelasticity: Admissibility and stability, Arch. Rational Mech. Anal., 97 (1987), pp. 353-394.
18. R. RACKE AND S. ZHENG, Global existence and asymptotic behavior in nonlinear thermoviscoelasticity, J. Differential Equations, 134 (1997), pp. 46-67.
19. W. SHEN AND S. ZHENG, On the coupled Cahn-Hilliard equations, Comm. Partial Differential Equations, 18 (1993), pp. 701-727.
20. J. SPREKELS AND S. ZHENG, Global solutions to the equations of a Ginzburg-Landau theory for structural phase transitions in shape memory alloys, Physica D, 39 (1989), pp. 59-76.
21. S. ZHENG, Nonlinear Parabolic Equations and Hyperbolic-Parabolic Coupled Systems, Pitman Series Monographs and Surveys in Pure and Applied Mathematics 76, Longman Group Limited, London, 1995.
22. P. ZHOU, Multiplicity of solutions to a nonlinear boundary value problem, J. Math. Anal. Appl., submitted.