On the finite dimension of attractors of doubly nonlinear parabolic systems with l-trajectories

El Ouardi, Hamid

Archivum mathematicum, Tome 43 (2007) no. 4, pp. 289-303

Abstract (VO)

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.

Détail

Classification : 35B40, 35B41, 35K50, 35K55, 35K57, 35K65, 37L30
Keywords: doubly nonlinear parabolic systems; existence of solutions; global and exponential attractor; fractal dimension and l-trajectories

Bibliographie

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

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

[3] 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. | MR

[4] 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. | MR | Zbl

[5] 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. | MR

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

[7] 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. | MR | Zbl

[8] 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. | MR

[9] 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. | MR

[10] El Ouardi H., de Thelin F.: Supersolutions and stabilization of the solutions of a nonlinear parabolic system. Publ. Mat. 33 (1989), 369–381. | MR

[11] 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

[12] 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 | Zbl

[13] 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 | Zbl

[14] 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

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

[16] Lions J. L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris 1969. | MR | Zbl

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

[18] Ladyzhznskaya O., Solonnikov V. A., Outraltseva N. N.: Linear and quasilinear equations of parabolic type. Trans. Amer. Math. Soc. XI, 1968.

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

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

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

[22] Foias C., Constantin P., Temam R.: Attractors representing turbulent flows. Mem. Am. Math. Soc. 314 (1985), 67p. | MR | Zbl

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

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

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

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

[27] Temam R.: Infinite dimensional dynamical systems in mechanics and physics. Appl. Math. Sci. 68, Springer-Verlag 1988. | MR | Zbl

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