Counting the set of equilibria for a one-dimensional full model for phase transitions with microscopic movements

Jiang, Jie; Zhang, Yanyan

doi:10.1090/S0033-569X-2012-01257-3

Authors: Jie Jiang and Yanyan Zhang
Journal: Quart. Appl. Math. 70 (2012), 665-683
MSC (2000): Primary 34B08, 35B40; Secondary 35K45, 80A22
DOI: https://doi.org/10.1090/S0033-569X-2012-01257-3
Published electronically: June 21, 2012
MathSciNet review: 3052084
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Abstract: This paper is concerned with the steady states of a one-dimensional strongly nonlinear PDE system arising from the study of phase transitions with microscopic movements. Using the plane analysis method, we prove that the number of equilibria is at most infinitely countable. Furthermore, we show that the global solution to the evolution system converges to an equilibrium in $H^3\times H^1$ as $t\rightarrow +\infty$.

References

V. Berti, M. Fabrizio, and C. Giorgi, Well-posedness for solid-liquid phase transitions with a fourth-order nonlinearity, Phys. D 236 (2007), no. 1, 13–21. MR 2385021, DOI https://doi.org/10.1016/j.physd.2007.07.009
Giovanna Bonfanti, Michel Frémond, and Fabio Luterotti, Global solution to a nonlinear system for irreversible phase changes, Adv. Math. Sci. Appl. 10 (2000), no. 1, 1–24. MR 1769184
Michel Frémond, Non-smooth thermomechanics, Springer-Verlag, Berlin, 2002. MR 1885252
M. Grinfeld and A. Novick-Cohen, Counting stationary solutions of the Cahn-Hilliard equation by transversality arguments, Proc. Roy. Soc. Edinburgh Sect. A 125 (1995), no. 2, 351–370. MR 1331565, DOI https://doi.org/10.1017/S0308210500028079

J. Jiang and B. Guo, Asymptotic behavior of solutions to a one-dimensional full model for phase transitions with microscopic movements, Discr. Cont. Dyn. Syst.-A, to appear.

Philippe Laurençot, Giulio Schimperna, and Ulisse Stefanelli, Global existence of a strong solution to the one-dimensional full model for irreversible phase transitions, J. Math. Anal. Appl. 271 (2002), no. 2, 426–442. MR 1923644, DOI https://doi.org/10.1016/S0022-247X%2802%2900127-0

S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels, Les Équations aux Dérivées Partielles (Paris, 1962) Éditions du Centre National de la Recherche Scientifique, Paris, 1963, pp. 87–89 (French). MR 0160856

S. Lojasiewicz, Ensembles semi-analytiques, Preprint, I.H.E.S., Bures-sur-Yvette, 1965.

Stanisław Łojasiewicz and Maria-A. Zurro, On the gradient inequality, Bull. Polish Acad. Sci. Math. 47 (1999), no. 2, 143–145. MR 1686676

F. Luterotti and U. Stefanelli, Existence result for the one-dimensional full model of phase transitions, Z. Anal. Anwendungen 21 (2002), no. 2, 335–350. MR 1915265, DOI https://doi.org/10.4171/ZAA/1081

F. Luterotti and U. Stefanelli, Errata and addendum to: “Existence result for the one-dimensional full model of phase transitions” [Z. Anal. Anwendungen 21 (2002), no. 2, 335–350; MR1915265 (2003e:80010)], Z. Anal. Anwendungen 22 (2003), no. 1, 239–240. MR 1962087, DOI https://doi.org/10.4171/ZAA/1143

Hiroshi Matano, Convergence of solutions of one-dimensional semilinear parabolic equations, J. Math. Kyoto Univ. 18 (1978), no. 2, 221–227. MR 501842, DOI https://doi.org/10.1215/kjm/1250522572

A. Novick-Cohen and M. Grinfeld, Counting the stationary states of the Sivashinsky equation, Nonlinear Anal. 24 (1995), no. 6, 875–881. MR 1320694, DOI https://doi.org/10.1016/0362-546X%2894%2900121-W

A. Novick-Cohen and L. A. Peletier, Steady states of the one-dimensional Cahn-Hilliard equation, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), no. 6, 1071–1098. MR 1263907, DOI https://doi.org/10.1017/S0308210500029747

A. Novick-Cohen and L. A. Peletier, The steady states of one-dimensional Sivashinsky equations, Quart. Appl. Math. 50 (1992), no. 4, 759–777. MR 1193665, DOI https://doi.org/10.1090/qam/1193665

A. Novick-Cohen and Songmu Zheng, The Penrose-Fife-type equations: counting the one-dimensional stationary solutions, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), no. 3, 483–504. MR 1396275, DOI https://doi.org/10.1017/S0308210500022873

Elisabetta Rocca and Riccarda Rossi, Global existence of strong solutions to the one-dimensional full model for phase transitions in thermoviscoelastic materials, Appl. Math. 53 (2008), no. 5, 485–520. MR 2469589, DOI https://doi.org/10.1007/s10492-008-0038-5

Renate Schaaf, Global solution branches of two-point boundary value problems, Lecture Notes in Mathematics, vol. 1458, Springer-Verlag, Berlin, 1990. MR 1090827

Renate Schaaf, Global behaviour of solution branches for some Neumann problems depending on one or several parameters, J. Reine Angew. Math. 346 (1984), 1–31. MR 727393, DOI https://doi.org/10.1515/crll.1984.346.1

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

Leon Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math. (2) 118 (1983), no. 3, 525–571. MR 727703, DOI https://doi.org/10.2307/2006981

Yanyan Zhang, Counting the stationary states and the convergence to equilibrium for the 1-D thin film equation, Nonlinear Anal. 71 (2009), no. 5-6, 1425–1437. MR 2524358, DOI https://doi.org/10.1016/j.na.2008.12.014

Yanyan Zhang, The steady states and convergence to equilibria for a 1-D chemotaxis model with volume-filling effect, Math. Methods Appl. Sci. 33 (2010), no. 1, 25–40. MR 2591221, DOI https://doi.org/10.1002/mma.1283

Zheng Songmu, Asymptotic behavior of solution to the Cahn-Hillard equation, Appl. Anal. 23 (1986), no. 3, 165–184. MR 870486, DOI https://doi.org/10.1080/00036818608839639

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

Songmu Zheng, Nonlinear evolution equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 133, Chapman & Hall/CRC, Boca Raton, FL, 2004. MR 2088362