Our aim in this paper is to study the long time behavior of a class of doubly nonlinear parabolic equations. In particular, we prove the existence of the global attractor which has, in one and two space dimensions, finite fractal dimension.
@article{bwmeta1.element.doi-10_1007_s11533-005-0010-5,
author = {Alain Miranville},
title = {Finite dimensional global attractor for a class of doubly nonlinear parabolic equations},
journal = {Open Mathematics},
volume = {4},
year = {2006},
pages = {163-182},
zbl = {1102.35023},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1007_s11533-005-0010-5}
}
Alain Miranville. Finite dimensional global attractor for a class of doubly nonlinear parabolic equations. Open Mathematics, Tome 4 (2006) pp. 163-182. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1007_s11533-005-0010-5/
[1] H.W. Alt and S. Luckhaus: “Quasilinear elliptic-parabolic differential equations”, Math. Z., Vol. 183, (1983), pp. 311–341. http://dx.doi.org/10.1007/BF01176474 | Zbl 0497.35049
[2] T. Arai: “On the existence of the solution for ∂ϕ(u′(t)) + ∂ψ(u(t)) ∋ f(t)”, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Vol. 26, (1979), pp. 75–96. | Zbl 0418.35056
[3] A.V. Babin and M.I. Vishik: Attractors of evolution equations, North-Holland, Amsterdam, 1992.
[4] A. Bamberger: “Etude d'une équation doublement non linéaire”, J. Funct. Anal., Vol. 24, (1977), pp. 148–155. http://dx.doi.org/10.1016/0022-1236(77)90051-9
[5] V. Barbu: Nonlinear semigroups and differential equations in Banach spaces, Noordhoff, Leiden, 1976.
[6] H. Brezis: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland, Amsterdam, 1973. | Zbl 0252.47055
[7] P. Colli: “On some doubly nonlinear evolution equations in Banach spaces”, Japan J. Indust. Appl. Math., Vol. 9, (1992), pp. 181–203. http://dx.doi.org/10.1007/BF03167565 | Zbl 0757.34051
[8] P. Colli and A. Visintin: “On a class of doubly nonlinear evolution problems”, Comm. Partial Diff. Eqns., Vol. 15, (1990), pp. 737–756. | Zbl 0707.34053
[9] E. DiBenedetto and R.E. Showalter: “Implicit degenerate evolution equations and applications”, S.I.A.M. J. Math. Anal., Vol. 12, (1981), pp. 731–751. http://dx.doi.org/10.1137/0512062 | Zbl 0477.47037
[10] A. Eden, C. Foias, B. Nicolaenko and R. Temam: Exponential attractors for dissipative evolution equations, Research in Applied Mathematics, Vol. 37, John-Wiley, New York, 1994. | Zbl 0842.58056
[11] A. Eden, B. Michaux and J.-M. Rakotoson: “Doubly nonlinear parabolic-type equations as dynamical systems”, J. Dyn. Diff. Eqns., Vol. 3, (1991), pp. 87–131. http://dx.doi.org/10.1007/BF01049490 | Zbl 0802.35011
[12] A. Eden and J.-M. Rakotoson: “Exponential attractors for a doubly nonlinear equation”, J. Math. Anal. Appl., Vol. 185(2), (1994), pp. 321–339. http://dx.doi.org/10.1006/jmaa.1994.1251
[13] M. Efendiev, A. Miranville and S. Zelik: “Exponential attractors for a nonlinear reaction-diffusion system in R 3”, C. R. Acad. Sci. Paris Sér. I, Vol. 330, (2000), pp. 713–718. | Zbl 1151.35315
[14] C.M. Elliott and R. Schätzle: “The limit of the anisotropic double-obstacle Allen-Cahn equation”, Proc. Royal Soc. Edin. A, Vol. 126, (1996), pp. 1217–1234. | Zbl 0865.35073
[15] O. Grange and F. Mignot: “Sur la résolution d'une équation et d'une inéquation paraboliques non linéaires”, J. Funct. Anal., Vol. 11, (1972), pp. 77–92. http://dx.doi.org/10.1016/0022-1236(72)90080-8 | Zbl 0251.35055
[16] M. Gurtin: “Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance”, Physica D, Vol. 92, (1996), pp. 178–192. http://dx.doi.org/10.1016/0167-2789(95)00173-5
[17] O.A. Ladyzhenskaya and N.N. Ural'ceva: Equations aux dérivées partielles de type elliptique, Monographies universitaires de Mathématiques, Vol. 31, Dunod, 1968.
[18] J.-L. Lions: Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1969.
[19] J. Malek and D. Prazak: “Long time behavior via the method of l-trajectories”, J. Diff. Eqns., Vol. 18(2), (2002), pp. 243–279. http://dx.doi.org/10.1006/jdeq.2001.4087
[20] D. Prazak: “A necessary and sufficient condition for the existence of an exponential attractor”, Cent. Eur. J. Math., Vol. 1(3), (2003), pp. 411–417.
[21] P.-A. Raviart: “Sur la résolution de certaines équations paraboliques non linéaires”, J. Funct. Anal., Vol. 5, (1970), pp. 299–328. http://dx.doi.org/10.1016/0022-1236(70)90031-5
[22] A. Segatti: “Global attractor for a class of doubly nonlinear abstract evolution equations”, Discrete Cont. Dyn. Systems, To appear. | Zbl 1092.37052
[23] K. Shirakawa: “Large time behavior for doubly nonlinear systems generated by sub-differentials”, Adv. Math. Sci. Appl., Vol. 10, (2000), pp. 77–92.
[24] R.E. Showalter: Monotone operators in Banach spaces and nonlinear partial differential equations, Amer. Math. Soc., Providence, R.I., 1997. | Zbl 0870.35004
[25] J. Simon: “Régularité de la solution d'un problème aux limites non linéaire”, Ann. Fac. Sci. Toulouse, Vol. 3(3–4), (1981), pp. 247–274. | Zbl 0487.35015
[26] J.E. Taylor and J.W. Cahn: “Linking anisotropic sharp and diffuse surface motion laws via gradient flows”, J. Statist. Phys., Vol. 77(1–2), (1993), pp. 183–197. | Zbl 0844.35044
[27] R. Temam: Infinite dimensional dynamical systems in mechanics and physics, 2nd ed., Springer-Verlag, 1997. | Zbl 0871.35001
[28] A. Visintin: Models of phase transitions, Birkhäuser, Boston, 1996.