ATTRACTORS FOR A CAGINALP PHASE-FIELD MODEL TYPE ON THE WHOLE SPACE R<sup>3</sup>

Brice Doumbé Bangola

doi:10.11948/2012018

2012 Volume 2 Issue 3

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2012 > 2(3): 251-272

Next Article Previous Article

Brice Doumbé Bangola. ATTRACTORS FOR A CAGINALP PHASE-FIELD MODEL TYPE ON THE WHOLE SPACE R3[J]. Journal of Applied Analysis & Computation, 2012, 2(3): 251-272. doi: 10.11948/2012018

Citation:

Brice Doumbé Bangola. ATTRACTORS FOR A CAGINALP PHASE-FIELD MODEL TYPE ON THE WHOLE SPACE R³[J]. Journal of Applied Analysis & Computation, 2012, 2(3): 251-272. doi: 10.11948/2012018

ATTRACTORS FOR A CAGINALP PHASE-FIELD MODEL TYPE ON THE WHOLE SPACE R³

Brice Doumbé Bangola

Laboratoire de Mathéatiques et Applications, Université de Poitiers, SP2MI, Boulevard Marie et Pierre Curie-Téléport 2, 86962 Chasseneuil Futuroscope Cedex, Poitiers, France

Abstract

Abstract

We consider in this paper a generalization of Caginalp phase-field system derived from a generalization of the Maxwell-Cattaneo law in an unbounded domain namely R³ in our case; which make the analysis challenging. We prove the well-posedness of the problem and the dissipativity of the associated semigroup. Finally, we study the long time behavior of solutions in terms of attractors.
- Caginalp system /
- Maxwell-Cattaneo law /
- well-posedness /
- dissipativity /
- long time behavior of solutions /
- global attractors
MSC: 35B41;35B45

FullText(HTML)

References

[1]	P. W. Bates and S. M. Zheng, Inertial manifolds and inetial sets for the phasefield equations, J. Dynam. Differential equations, 4(1992), 375-398. Google Scholar
[2]	D. Brochet, D. Hilhorst and A. Novick-Cohen, Finite-dimensional exponential attractor for a model for order-desorder and phase separation, Appl. Math. Lett., 7(1992), 83-87. Google Scholar
[3]	D. Brochet and D. Hilhorst, Universal attractor and inertial sets for the phasefield model, Appl. Math. Lett., 4(1991), 59-63. Google Scholar
[4]	D. Brochet, D. Hilhorst and X. Chen, Finite-dimensional exponential attractor for the phase field model, Appl. Anal., 49(1993),197-212. Google Scholar
[5]	A. V. Babin and M. I. Vishik, Attractors of partial differential evolution equations in unbouded domain, Proc. Roy. Soc. Edinburgh Sect. A, 116(1990), 221-243. Google Scholar
[6]	V. Belleri and V. Pata, Attractors for semilinear strogly damped wave equations on R³, Discrete Contin. Dynam. Systems., 7(2001), 719-735. Google Scholar
[7]	V. Belleri, Analisi asintoticadi equazioni delle onde fortamente smortzate in uno dominio illimitato, Master thesis, Politecnico di Milano, Italia. April 2005. Google Scholar
[8]	A. V. Babin and M. I. Vishik, Attractors for evolution equations, volume 25 of studies in Mathematics ans its Applications, North-Holland, Publishing Co., Amsterdam, 1992. Translated ans revised from 1989 Russian original by Babin. Google Scholar
[9]	B. D. Bangola, Etude de modèles de Champs de phase de type Caginalp, Ph.D. thesis, Université de Poitiers, France. to appear. Google Scholar
[10]	G. Caginalp, An analysis of a phase-field model of a free boundary, Arch. Rational Mech. Anal., 92(1986), 205-245. Google Scholar
[11]	L. Cherfils and A. Miranville, On the Caginalp system with dynamics boundary conditions and singular potentials, Appl. Math., 54(2009), 89-115. Google Scholar
[12]	M. Conti and G. Mola, Attractors for a phase field model on R³, Adv. Math. Sci. Appl., 15(2005), 527-543. Google Scholar
[13]	M. Conti, V. Pata and M. Squassima, Strongly damped wave equations on R³ with critical nonlinearities, Commun. Appl. Anal., 9(2005), 161-176. Google Scholar
[14]	A. Eden, C. Foias, B. Nicolenko, and R. Temam, Exponential attractors for dissipative evolution equations, volume 37 of RAM:Research in Applied Mathematics. Msson, Paris, 1994. Google Scholar
[15]	J. K. Hale, Asyptotic behavior of dissipative systems, volume 25 of Mathematical surveys and Monographs, Mathematical American Society, Providence, RI, 1988. Google Scholar
[16]	A. Miranville and R. Quintanilla, A generalization of the Caginalp phase-field system based on the Maxwell-Cattaneo law, Nonlinear Anal., 71(2009), 2278-2298. Google Scholar
[17]	A. Miranville and R. Quintanilla, A Caginalp phase-field system with nonlinear coupling, Nonlinear Anal. Real World Appl., 11(2010), 2849-2861. Google Scholar
[18]	A. Miranville and R. Quintanilla, Some generaizations of the Caginalp phasefield system, Appl. Anal., 88(2009), 897-894. Google Scholar
[19]	A. Miranville and R. Quintanilla, A phase-field model based on the three-phaselag heat conduction, Appl. Math. Optim., 63(2011), 133-150. Google Scholar
[20]	F. Morillas and J. Valero, Asymptotic compactness and attractors for phasefield equation in R³, Set-Valued Anal., 16(2008), 861-897. Google Scholar
[21]	A. Miranville and S. Zelik, Doubly nonlinear Cahn-Hilliard-Gurtin equations, Hokkaido Math. J., 38(2009), 315-360. Google Scholar
[22]	A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, In Handbook of differential equations:evolutionary equations. Vol. IV, Handb. Diff. Equ., 4(2008), 103-200. Google Scholar
[23]	V. Pata, Attractors for a damped wave equation on R³ with linear memory, Math. Methods Appl. Sci., 23(2000), 633-653. Google Scholar
[24]	R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, volume 68 of Applied Mathematical Sciences, Springer-Verlag, New York, second edition, 1997. Google Scholar
[25]	S. V. Zelik, The attractor for a nonlinear hyperbolic equation in the unbounded domain. Discrete Contin, Dynam. Systemes, 7(2001), 593-641. Google Scholar