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Wehbe, Charbel

On a Caginalp phase-field system with a logarithmic nonlinearity. (English) Zbl 1363.35045

Appl. Math., Praha 60, No. 4, 355-382 (2015).
Summary: We consider a phase field system based on the Maxwell Cattaneo heat conduction law, with a logarithmic nonlinearity, associated with Dirichlet boundary conditions. In particular, we prove, in one and two space dimensions, the existence of a solution which is strictly separated from the singularities of the nonlinear term and that the problem possesses a finite-dimensional global attractor in terms of exponential attractors.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35B41 Attractors
35K51 Initial-boundary value problems for second-order parabolic systems
80A22 Stefan problems, phase changes, etc.
80A20 Heat and mass transfer, heat flow (MSC2010)
35Q53 KdV equations (Korteweg-de Vries equations)
45K05 Integro-partial differential equations
35K55 Nonlinear parabolic equations
35G30 Boundary value problems for nonlinear higher-order PDEs
92D50 Animal behavior

Keywords:

Caginalp phase-field system; Dirichlet boundary conditions; well-posedness; long time behavior of solution; global attractor; exponential attractor; Maxwell-Cattaneo law; logarithmic potential
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\textit{C. Wehbe}, Appl. Math., Praha 60, No. 4, 355--382 (2015; Zbl 1363.35045)
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References:

[1] Babin, A.; Nicolaenko, B., Exponential attractors for reaction-diffusion systems in an unbounded domain, J. Dyn. Differ. Equations, 7, 567-590, (1995) · Zbl 0846.35061
[2] Brochet, D.; Hilhorst, D., Universal attractor and inertial sets for the phase field model, Appl. Math. Lett., 4, 59-62, (1991) · Zbl 0773.35028
[3] Brochet, D.; Hilhorst, D.; Chen, X., Finite dimensional exponential attractor for the phase field model, Appl. Anal., 49, 197-212, (1993) · Zbl 0790.35052
[4] Brochet, D.; Hilhorst, D.; Novick-Cohen, A., Finite-dimensional exponential attractor for a model for order-disorder and phase separation, Appl. Math. Lett., 7, 83-87, (1994) · Zbl 0803.35076
[5] Caginalp, G., An analysis of a phase field model of a free boundary, Arch. Ration. Mech. Anal., 92, 205-245, (1986) · Zbl 0608.35080
[6] Caginalp, G., The role of microscopic anisotropy in the macroscopic behavior of a phase boundary, Ann. Phys., 172, 136-155, (1986) · Zbl 0639.58038
[7] Cherfils, L.; Miranville, A., Some results on the asymptotic behavior of the Caginalp system with singular potentials, Adv. Math. Sci. Appl., 17, 107-129, (2007) · Zbl 1145.35042
[8] Conti, M.; Gatti, S.; Miranville, A., Asymptotic behavior of the Caginalp phase-field system with coupled dynamic boundary conditions, Discrete Contin. Dyn. Syst., Ser. S, 5, 485-505, (2012) · Zbl 1244.35067
[9] Fabrie, P.; Galusinski, C., Exponential attractors for partially dissipative reaction system, Asymptotic Anal., 12, 329-354, (1996) · Zbl 1028.35026
[10] Gajewski, H.; Zacharias, K., Global behaviour of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195, 77-114, (1998) · Zbl 0918.35064
[11] Grasselli, M.; Miranville, A.; Pata, V.; Zelik, S., Well-posedness and long time behavior of a parabolic-hyperbolic phase-field system with singular potentials, Math. Nachr., 280, 1475-1509, (2007) · Zbl 1133.35017
[12] A. Kufner, O. John, S. Fučík: Function Spaces. Monographs and Textsbooks on Mechanics of Solids and Fluids. Mechanics: Analysis. Noordhoff International Publishing, Leyden, Academia, Prague, 1977.
[13] L. D. Landau, E. M. Lifschitz: Course of Theoretical Physics. Vol. 1. Mechanics. Akademie, Berlin, 1981. (In German.)
[14] Miranville, A., Exponential attractors for a class of evolution equations by a decomposition method, C. R. Acad. Sci., Paris, Sér. I, Math., 328, 145-150, (1999) · Zbl 1101.35334
[15] Miranville, A., On a phase-field model with a logarithmic nonlinearity, Appl. Math., Praha, 57, 215-229, (2012) · Zbl 1265.35139
[16] Miranville, A., Some mathematical models in phase transition, Discrete Contin. Dyn. Syst., Ser. S, 7, 271-306, (2014) · Zbl 1275.35048
[17] Miranville, A.; Quintanilla, R., A generalization of the Caginalp phase-field system based on the Cattaneo law, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, 71, 2278-2290, (2009) · Zbl 1167.35304
[18] Miranville, A.; Quintanilla, R., Some generalizations of the Caginalp phase-field system, Appl. Anal., 88, 877-894, (2009) · Zbl 1178.35194
[19] Miranville, A.; Zelik, S., Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math. Methods Appl. Sci., 27, 545-582, (2004) · Zbl 1050.35113
[20] A. Miranville, S. Zelik: Attractors for dissipative partial differential equations in bounded and unbounded domains. Handbook of Differential Equations: Evolutionary Equations. Vol. IV (C. M. Dafermos et al., eds.). Elsevier/North-Holland, Amsterdam, 2008, pp. 103-200. · Zbl 1221.37158
[21] R. Temam: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Applied Mathematical Sciences 68, Springer, New York, 1997. · Zbl 0871.35001
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.