On the finite dimension of attractors of doubly nonlinear parabolic systems with l-trajectories
El Ouardi, Hamid
Archivum Mathematicum, Tome 043 (2007), p. 289-303 / Harvested from Czech Digital Mathematics Library

This paper is concerned with the asymptotic behaviour of a class of doubly nonlinear parabolic systems. In particular, we prove the existence of the global attractor which has, in one and two space dimensions, finite fractal dimension.

Publié le : 2007-01-01
Classification:  35B40,  35B41,  35K50,  35K55,  35K57,  35K65,  37L30
@article{108073,
     author = {Hamid El Ouardi},
     title = {On the finite dimension of attractors of doubly nonlinear parabolic systems with l-trajectories},
     journal = {Archivum Mathematicum},
     volume = {043},
     year = {2007},
     pages = {289-303},
     zbl = {1164.35045},
     mrnumber = {2378529},
     language = {en},
     url = {http://dml.mathdoc.fr/item/108073}
}
El Ouardi, Hamid. On the finite dimension of attractors of doubly nonlinear parabolic systems with l-trajectories. Archivum Mathematicum, Tome 043 (2007) pp. 289-303. http://gdmltest.u-ga.fr/item/108073/

Alt H. W.; Luckhaus S. Quasilinear elliptic and parabolic differential equations, Math. Z. 183 (1983), 311–341. (1983) | MR 0706391

Bamberger A. Etude d’une équation doublement non linéaire, J. Funct. Anal. 24 (1977), 148–155. (1977) | MR 0470490 | Zbl 0345.35059

Blanchard D.; Francfort G. Study of doubly nonlinear heat equation with no growth assumptions on the parabolic term, Siam. J. Anal. Math. 9 (5), (1988), 1032–1056. (1988) | MR 0957665

Diaz J.; De Thelin F. On a nonlinear parabolic problem arising in some models related to turbulent flows, Siam. J. Anal. Math. 25 (4), (1994), 1085–1111. (1994) | MR 1278892 | Zbl 0808.35066

Diaz J.; Padial J. F. Uniqueness and existence of solutions in the BV$_{t}(Q)$ space to a doubly nonlinear parabolic problem, Publ. Mat. 40 (1996), 527–560. (1996) | MR 1425634

Eden A.; Michaux B.; Rakotoson J. M. Doubly nonlinear parabolic type equations as dynamical systems, J. Dynam. Differential Equations 3 (1), (1991), 87–131. (1991) | MR 1094725 | Zbl 0802.35011

Eden A.; Michaux B.; Rakotoson J. M. Semi-discretized nonlinear evolution equations as discrete dynamical systems and error analysis, Indiana Univ. Math. J. 39 (3), (1990), 737–783. (1990) | MR 1078736 | Zbl 0732.35037

El Hachimi A.; De Thelin F. Supersolutions and stabilisation of the solutions of the equation $\ u_{t}-\triangle _{p}u=f(x,u)$, Part I, Nonlinear Anal., Theory Methods Appl. 12 (12), (1988), 1385–1398. (1988) | MR 0972407

El Hachimi A.; De Thelin F. Supersolutions and stabilisation of the solutions of the equation $u_{t}-\triangle _{p}u=f(x,u)$, Part II, Publ. Mat. 35 (1991), 347–362. (1991) | MR 1201561

El Ouardi H.; De Thelin F. Supersolutions and stabilization of the solutions of a nonlinear parabolic system, Publ. Mat. 33 (1989), 369–381. (1989) | MR 1030974

El Ouardi H.; El Hachimi A. Existence and attractors of solutions for nonlinear parabolic systems, Electron. J. Qual. Theory Differ. Equ. 5 (2001), 1–16. | MR 1873359

El Ouardi H.; El Hachimi A. Existence and regularity of a global attractor for doubly nonlinear parabolic equations, Electron. J. Differ. Equ. (45) (2002), 1–15. | MR 1907721 | Zbl 0993.35021

El Ouardi H.; El Hachimi A. Attractors for a class of doubly nonlinear parabolic systems, Electron. J. Qual. Theory Differ. Equ. (1) (2006), 1–15. | MR 2210533 | Zbl 1085.35064

Dung L. Ultimate boundedness of solutions and gradients of a class of degenerate parabolic systems, Nonlinear Anal., Theory Methods Appl. 39 (2000), 157–171. | MR 1722106

Dung L. Global attractors and steady states for a class of reaction diffusion systems, J. Differential Equations 147 (1998), 1–29. (1998) | MR 1632736

Lions J. L. Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris 1969. (1969) | MR 0259693 | Zbl 0189.40603

Langlais M.; Phillips D. Stabilization of solution of nonlinear and degenerate evolution equatins, Nonlinear Anal., Theory Methods Appl. 9 (1985), 321–333. (1985) | MR 0783581

Ladyzhznskaya O.; Solonnikov V. A.; Outraltseva N. N. Linear and quasilinear equations of parabolic type, Trans. Amer. Math. Soc. XI, 1968. (1968)

Marion M. Attractors for reaction-diffusion equation: existence and estimate of their dimension, Appl. Anal. 25 (1987), 101–147. (1987) | MR 0911962

Málek J.; Pražák D. Long time behaviour via the method of l-trajectories, J. Differential Equations 18 (2), (2002), 243–279.

Miranville A. Finite dimensional global attractor for a class of doubly nonlinear parabolic equations, Central European J. Math. 4 (1), (2006), 163–182. | MR 2213032 | Zbl 1102.35023

Foias C.; Constantin P.; Temam R. Attractors representing turbulent flows, Mem. Am. Math. Soc. 314 (1985), 67p. (1985) | MR 0776345 | Zbl 0567.35070

Foias C.; Temam R. Structure of the set of stationary solutions of the Navier-Stokes equations, Commun. Pure Appl. Math. 30 (1977), 149–164. (1977) | MR 0435626 | Zbl 0335.35077

Simon J. Régularité de la solution d’un problème aux limites non linéaire, Ann. Fac. Sci. Toulouse Math. (6),(1981), 247–274. (1981) | MR 0658735

Schatzman M. Stationary solutions and asymptotic behavior of a quasilinear parabolic equation, Indiana Univ. Math. J. 33 (1984), 1–29. (1984) | MR 0726104

Raviart P. A. Sur la résolution de certaines équations paraboliques non linéaires, J. Funct. Anal. 5 (1970), 299–328. (1970) | MR 0257585 | Zbl 0199.42401

Temam R. Infinite dimensional dynamical systems in mechanics and physics, Appl. Math. Sci. 68, Springer-Verlag 1988. (1988) | MR 0953967 | Zbl 0662.35001

Tsutsumi M. On solutions of some doubly nonlinear degenerate parabolic systems with absorption, J. Math. Anal. Appl. 132 (1988), 187–212. (1988) | MR 0942364