Cholewa, Czaja, Mola: Remarks on the Fractal Dimension of Bi-Space Global and Exponential Attractors

Cholewa, Jan W. and Czaja, Radoslaw and Mola, Gianluca:
Remarks on the Fractal Dimension of Bi-Space Global and Exponential Attractors
Bollettino dell'Unione Matematica Italiana Serie 9 1 (2008), fasc. n.1, p. 121-145, (English)
pdf (490 Kb), djvu (249 Kb). | MR 2388001 | Zbl 1213.37111

Sunto

In questo lavoro sono considerate le nozioni di attrattori globali ed esponenziali "bi-space" per sistemi dinamici continui, e discusse limitazioni relative alla loro dimensione frattale in spazi di funzioni opportuni. Di particolare interesse è il caso in cui il sistema presenta un parziale effetto regolarizzante, ed alcuni esempi con questa proprietà sono mostrati.

Referenze Bibliografiche

[1] J. M. ARRIETA - J. W. CHOLEWA - T. DLOTKO - A. RODRIÂGUEZ-BERNAL, Dissipative parabolic equations in locally uniform spaces, Math. Nachr., 280 (2007), 1643-1663. | fulltext (doi) | MR 2351571 | Zbl 1146.35015

[2] H. AMANN, Global existence for semilinear parabolic systems, J. Reine Angew. Math., 360 (1985), 47-83. | fulltext EuDML | fulltext (doi) | MR 799657 | Zbl 0564.35060

[3] A. V. BABIN - M. I. VISHIK, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. | MR 1156492 | Zbl 0778.58002

[4] A. N. CARVALHO - J. W. CHOLEWA, Local well posedness for strongly damped wave equations with critical nonlinearities, Bulletin of the Australian Mathematical Society, 66 (2002), 443-463. | fulltext (doi) | MR 1939206 | Zbl 1020.35059

[5] A. N. CARVALHO - J. W. CHOLEWA, Attractors for strongly damped wave equations with critical nonlinearities, Pacific Journal of Mathematics, 207 (2002), 287-310. | fulltext (doi) | MR 1972247 | Zbl 1060.35082

[6] A. N. CARVALHO - J. W. CHOLEWA, Continuation and asymptotics of solutions to semilinear parabolic equations with critical nonlinearities, J. Math. Anal. Appl., 310 (2005), 557-578. | fulltext (doi) | MR 2022944 | Zbl 1077.35031

[7] S. CHEN - R. TRIGGIANI, Proof of two conjectures on structural damping for elastic systems: The case a=1/2, Lecture Notes in Mathematics 1354, Springer, 1988, 234-256. | fulltext (doi) | MR 996678

[8] J. W. CHOLEWA - T. DLOTKO, Global Attractors in Abstract Parabolic Problems, Cambridge University Press, Cambridge, 2000. | fulltext (doi) | MR 1778284 | Zbl 0954.35002

[9] M. CONTI - V. PATA - M. SQUASSINA, Singular limit of differential systems with memory, Indiana Univ. Math. J., 55 (2006), 169-215. | fulltext (doi) | MR 2207550 | Zbl 1100.35018

[10] L. DUNG - B. NICOLAENKO, Exponential attractors in Banach spaces, J. Dynam. Differential Equations, 13 (2001), 791-806. | fulltext (doi) | MR 1860286 | Zbl 1040.37069

[11] A. EDEN - C. FOIAS - B. NICOLAENKO - R. TEMAM, Exponential Attractors for Dissipative Evolution Equations, John Wiley and Sons, Ltd., Chichester, 1994. | MR 1335230 | Zbl 0842.58056

[12] M. EFENDIEV - A. MIRANVILLE - S. ZELIK, Exponential attractors for a nonlinear reaction-diffusion system in $\mathbb{R}^3$ , C. R. Acad. Sci. Paris Sr. I Math., 330 (2000), 713- 718. | fulltext (doi) | MR 1763916 | Zbl 1151.35315

[13] P. FABRIE - C. GALUSINSKI - A. MIRANVILLE - S. ZELIK, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Cont. Dyn. Systems, 10 (2004), 211-238. | fulltext (doi) | MR 2026192 | Zbl 1060.35011

[14] S. GATTI - M. GRASSELLI - A. MIRANVILLE - V. PATA, A construction of a robust family of exponential attractors, Proc. Amer. Math. Soc., 134 (2006), 117-127. | fulltext (doi) | MR 2170551 | Zbl 1078.37047

[15] S. GATTI - M. GRASSELLI - V. PATA, Exponential attractors for a phase-field model with memory and quadratic nonlinearities, Indiana Univ. Math. J., 53 (2004), 719- 754. | fulltext (doi) | MR 2086698 | Zbl 1070.37056

[16] J. K. HALE, Asymptotic Behavior of Dissipative Systems, AMS, Providence, R.I., 1988. | MR 941371

[17] D. HENRY, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 1981. | MR 610244 | Zbl 0456.35001

[18] O. A. LADYŽENSKAYA, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991.

[19] O. A. LADYŽENSKAYA - V. A. SOLONNIKOV - N. N. URAL'CEVA, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, AMS, Providence, R.I., 1967. | MR 241822

[20] DE-SHENG LI - CHEN-KUI ZHONG, Global attractor for the Cahn-Hilliard system with fast growing nonlinearity, J. Diff. Equations, 149 (1998), 191-210. | fulltext (doi) | MR 1646238 | Zbl 0912.35029

[21] J. MÁLEK - D. PRAŽAK, Large time behavior via the method of $\ell$-trajectories, J. Diff. Equations, 181 (2002), 243-279. | fulltext (doi) | MR 1907143

[22] G. MOLA, Global attractors for a three-dimensional conserved phase-field system with memory, Commun. Pure Appl. Anal., to appear. | fulltext (doi) | MR 2373219 | Zbl 1144.35354

[23] G. MOLA, Stability of global and exponential attractors for a three-dimensional conserved phase-field system with memory, submitted. | fulltext (doi) | MR 2373219 | Zbl 1185.35027

[24] V. PATA - M. SQUASSINA, On the strongly damped wave equation, Comm. Math. Phys., 253 (2005), 511-533. | fulltext (doi) | MR 2116726 | Zbl 1068.35077

[25] V. PATA - S. ZELIK, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 1495-1506. | fulltext (doi) | MR 2229785 | Zbl 1113.35023

[26] V. PATA - S. ZELIK, A result on the existence of global attractors for semigroups of closed operators, Commun. Pure Appl. Anal., 6 (2007), 481-486. | fulltext (doi) | MR 2289833 | Zbl 1152.47046

[27] V. PATA - A. ZUCCHI, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529. | MR 1907454 | Zbl 0999.35014

[28] P. POLÁČIK, Parabolic equations: asymptotic behavior and dynamics on invariant manifolds, in: Handbook of Dynamical Systems Vol. 2, North-Holland, Amsterdam, 2002, 835-883.

[29] R. TEMAM, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1988. | fulltext (doi) | MR 953967 | Zbl 0662.35001

[30] H. TRIEBEL, Interpolation Theory, Function Spaces, Differential Operators, Veb Deutscher, Berlin, 1978. | MR 500580

[31] W. VON WAHL, Global solutions to evolution equations of parabolic type, in: Differential Equations in Banach Spaces, Proceedings, 1985, Springer-Verlag, Berlin, 1986, 254-266. | fulltext (doi) | MR 872532

[32] G. F. WEBB, Existence and asymptotic behavior for a strongly damped nonlinear wave equation, Can. J. Math., 32 (1980), 631-643. | fulltext (doi) | MR 586981 | Zbl 0414.35046

bdim: Biblioteca Digitale Italiana di Matematica

Un progetto SIMAI e UMI

Referenza completa

Sunto

Referenze Bibliografiche