Counting the set of equilibria for a one-dimensional full model for phase transitions with microscopic movements. (English) Zbl 1331.35164
The number of steady states for a one-dimensional strongly nonlinear PDE system arising from the study of phase transition with microscopic movements is investigated in the paper. The existence of an infinitely countable number of solutions for the associated stationary problem of this system with respect to parameter values is also pointed out. A detailed classification of the level set for the time maps is given.
Reviewer: Ioana Dragomirescu (Stuttgart)
MSC:
| 35K45 | Initial value problems for second-order parabolic systems |
| 34B08 | Parameter dependent boundary value problems for ordinary differential equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 80A22 | Stefan problems, phase changes, etc. |
| 82C26 | Dynamic and nonequilibrium phase transitions (general) in statistical mechanics |
References:
| [1] | 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. · Zbl 1128.35354 · doi:10.1016/j.physd.2007.07.009 |
| [2] | 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. · Zbl 0956.35122 |
| [3] | Michel Frémond, Non-smooth thermomechanics, Springer-Verlag, Berlin, 2002. · Zbl 0990.80001 |
| [4] | 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. · Zbl 0828.34007 · doi:10.1017/S0308210500028079 |
| [5] | 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. · Zbl 1235.35039 |
| [6] | 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. · Zbl 1004.35058 · doi:10.1016/S0022-247X(02)00127-0 |
| [7] | 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). |
| [8] | S. Lojasiewicz, Ensembles semi-analytiques, Preprint, I.H.E.S., Bures-sur-Yvette, 1965. |
| [9] | Stanisław Łojasiewicz and Maria-A. Zurro, On the gradient inequality, Bull. Polish Acad. Sci. Math. 47 (1999), no. 2, 143 – 145. · Zbl 0932.32009 |
| [10] | 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. · Zbl 1003.80003 · doi:10.4171/ZAA/1081 |
| [11] | 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. · Zbl 1003.80003 · doi:10.4171/ZAA/1143 |
| [12] | Hiroshi Matano, Convergence of solutions of one-dimensional semilinear parabolic equations, J. Math. Kyoto Univ. 18 (1978), no. 2, 221 – 227. · Zbl 0387.35008 |
| [13] | A. Novick-Cohen and M. Grinfeld, Counting the stationary states of the Sivashinsky equation, Nonlinear Anal. 24 (1995), no. 6, 875 – 881. · Zbl 0922.35058 · doi:10.1016/0362-546X(94)00121-W |
| [14] | 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. · Zbl 0818.35127 · doi:10.1017/S0308210500029747 |
| [15] | A. Novick-Cohen and L. A. Peletier, The steady states of one-dimensional Sivashinsky equations, Quart. Appl. Math. 50 (1992), no. 4, 759 – 777. · Zbl 0764.35094 · doi:10.1090/qam/1193665 |
| [16] | 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. · Zbl 0857.35055 · doi:10.1017/S0308210500022873 |
| [17] | 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. · Zbl 1177.76018 · doi:10.1007/s10492-008-0038-5 |
| [18] | Renate Schaaf, Global solution branches of two-point boundary value problems, Lecture Notes in Mathematics, vol. 1458, Springer-Verlag, Berlin, 1990. · Zbl 0780.34010 |
| [19] | 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. · Zbl 0513.34033 · doi:10.1515/crll.1984.346.1 |
| [20] | Wei Xi Shen and Song Mu Zheng, On the coupled Cahn-Hilliard equations, Comm. Partial Differential Equations 18 (1993), no. 3-4, 701 – 727. · Zbl 0815.35041 · doi:10.1080/03605309308820946 |
| [21] | 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. · Zbl 0549.35071 · doi:10.2307/2006981 |
| [22] | 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. · Zbl 1198.35134 · doi:10.1016/j.na.2008.12.014 |
| [23] | 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. · Zbl 1196.65156 · doi:10.1002/mma.1283 |
| [24] | Zheng Songmu, Asymptotic behavior of solution to the Cahn-Hillard equation, Appl. Anal. 23 (1986), no. 3, 165 – 184. · Zbl 0582.34070 · doi:10.1080/00036818608839639 |
| [25] | 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. · Zbl 0835.35003 |
| [26] | 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. · Zbl 1085.47058 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
