Citation: Armel Andami Ovono, Alain Miranville. On the Caginalp phase-field system based on the Cattaneo law with nonlinear coupling[J]. AIMS Mathematics, 2016, 1(1): 24-42. doi: 10.3934/Math.2016.1.24
[1] | Caginalp G (1986) An analysis of a phase field model of a free boundary. Arch Ration Mech Anal ,92: 205-245. |
[2] | Aizicovici S, Feireisl E (2001) Long-time stabilization of solutions to a phase-field model with memory. J Evol Equ ,1: 69-84. |
[3] | Aizicovici S, Feireisl E (2001) Long-time convergence of solutions to a phase-field system. Math Methods Appl Sci ,24: 277-287. |
[4] | Brochet D, Chen X, Hilhorst D (1993) Finite dimensional exponential attractors for the phase-field model. Appl Anal ,49: 197-212. |
[5] | M. Brokate, J. Sprekels, Hysteresis and Phase Transitions, Springer, New York, 1996. |
[6] | Cherfils L, Miranville A (2007) Some results on the asymptotic behavior of the Caginalp system with singular potentials. Adv Math Sci Appl . |
[7] | Cherfils L, Miranville A (2009) On the Caginalp system with dynamic boundary conditions and singular potentials. Appl Math ,54: 89-115. |
[8] | Chill R, Fasangov E′a, J. Pr¨uss (2006) Convergence to steady states of solutions of the Cahn-Hilliard equation with dynamic boundary conditions. Math Nachr ,279: 1448-1462. |
[9] | C.I. Christov, P.M. Jordan, Heat conduction paradox involving second-sound propagation in moving media, Phys. Rev. Lett., 94 (2005), 154-301. |
[10] | J.N. Flavin, R.J. Knops, and L.E. Payne, Decay estimates for the constrained elastic cylinder of variable cross-section, Quart. Appl. Math., 47 (1989), 325-350. |
[11] | Gatti S, Miranville A (2006) Asymptotic behavior of a phase-field system with dynamic boundary conditions, in: Di erential Equations: Inverse and Direct Problems (Proceedings of the workshop “Evolution Equations: Inverse and Direct Problems ”, Cortona, June 21-25, 2004), in A. Favini, A. Lorenzi (Eds), A Series of Lecture Notes in Pure and Applied Mathematics ,251: 149-170. |
[12] | C. Giorgi, M. Grasselli, and V. Pata, Uniform attractors for a phase-field model with memory and quadratic nonlinearity, Indiana Univ. Math. J., 48 (1999), 1395-1446. |
[13] | Grasseli M, Miranville A, Pata V, Zelik S (2007) Well-posedness and long time behavior of a parabolic-hyperbolic phase-field system with singular potentials. Math Nachr ,280: 1475-1509. |
[14] | M. Grasselli, On the large time behavior of a phase-field system with memory, Asymptot. Anal., 56 (2008), 229-249. |
[15] | M. Grasselli, V. Pata, Robust exponential attractors for a phase-field system with memory J. Evol. Equ., 5 (2005), 465-483. |
[16] | M. Grasselli, H. Petzeltová, and G. Schimperna, Long time behavior of solutions to the Caginalp system with singular potentials, Z. Anal. Anwend., 25 (2006), 51-73. |
[17] | M. Grasselli, H. Wu, and S. Zheng, Asymptotic behavior of a non-isothermal Ginzburg-Landau model, Quart. Appl. Math., 66 (2008), 743-770. |
[18] | A.E. Green, P.M. Naghdi, A new thermoviscous theory for fluids, J. Non-Newtonian Fluid Mech., 56 (1995), 289-306. |
[19] | A.E. Green, P.M. Naghdi, A re-examination of the basic postulates of thermomechanics, Proc. Roy. Soc. Lond. A., 432 (1991), 171-194. |
[20] | A.E. Green, P.M. Naghdi, On undamped heat waves in an elastic solid, J. Thermal. Stresses, 15 (1992), 253-264. |
[21] | J. Jiang, Convergence to equilibrium for a parabolic-hyperbolic phase-field model with Cattaneo heat flux law, J. Math. Anal. Appl., 341 (2008), 149-169. |
[22] | J. Jiang, Convergence to equilibrium for a fully hyperbolic phase field model with Cattaneo heat flux law, Math. Methods Appl. Sci., 32 (2009), 1156-1182. |
[23] | Ph. Laurençot, Long-time behaviour for a model of phase-field type, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 167-185. |
[24] | A. Miranville, R. Quintanilla, Some generalizations of the Caginalp phase-field system, Appl. Anal., 88 (2009), 877-894. |
[25] | A. Miranville, R. Quintanilla, A generalization of the Caginalp phase-field system based on the Cattaneo law, Nonlinear Anal. TMA., 71 (2009), 2278-2290 |
[26] | A. Miranville, R. Quintanilla, A Caginalp phase-field system with a nonlinear coupling. Nonlinear Anal.: Real World Applications, 11 (2010), 2849-2861. |
[27] | A. Miranville, S. Zelik, Robust exponential attractors for singularly perturbed phase-field type equations, Electron. J. Diff. Equ., (2002), 1-28. |
[28] | A. Miranville, S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in: C.M. Dafermos, M. Pokorny (Eds.) Handbook of Differential Equations, Evolutionary Partial Differential Equations. Elsevier, Amsterdam, 2008. |
[29] | A. Novick-Cohen, A phase field system with memory: Global existence, J. Int. Equ. Appl. 14 (2002), 73-107. |
[30] | R. Quintanilla, On existence in thermoelasticity without energy dissipation, J. Thermal. Stresses, 25 (2002), 195-202. |
[31] | R. Quintanilla, End effects in thermoelasticity, Math. Methods Appl. Sci.. 24 (2001), 93-102. |
[32] | R. Quintanilla, R. Racke, Stability in thermoelasticity of type Ⅲ, Discrete Contin. Dyn. Syst. B, 3 (2003), 383-400. |
[33] | R. Quintanilla, Phragmén-Lindelöf alternative for linear equations of the anti-plane shear dynamic problem in viscoelasticity, Dynam. Contin. Discrete Impuls. Systems, 2 (1996), 423-435. |
[34] | R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, second edition, Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1997. |
[35] | Z. Zhang, Asymptotic behavior of solutions to the phase-field equations with Neumann boundary conditions, Comm. Pure Appl. Anal., 4 (2005), 683-693. |