Remarks on the Fractal Dimension of Bi-Space Global and Exponential Attractors

 Access to full text

 Full (PDF)

 Full (DjVu)

1. ARRIETA, J. M. - CHOLEWA, J. W. - DLOTKO, T. - RODRIÂGUEZ-BERNAL, A., Dissipative parabolic equations in locally uniform spaces, Math. Nachr., 280 (2007), 1643-1663. Zbl1146.35015MR2351571DOI10.1002/mana.200510569
2. AMANN, H., Global existence for semilinear parabolic systems, J. Reine Angew. Math., 360 (1985), 47-83. Zbl0564.35060MR799657DOI10.1515/crll.1985.360.47
3. BABIN, A. V. - VISHIK, M. I., Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. Zbl0778.58002MR1156492
4. CARVALHO, A. N. - CHOLEWA, J. W., Local well posedness for strongly damped wave equations with critical nonlinearities, Bulletin of the Australian Mathematical Society, 66 (2002), 443-463. Zbl1020.35059MR1939206DOI10.1017/S0004972700040296
5. CARVALHO, A. N. - CHOLEWA, J. W., Attractors for strongly damped wave equations with critical nonlinearities, Pacific Journal of Mathematics, 207 (2002), 287-310. Zbl1060.35082MR1972247DOI10.2140/pjm.2002.207.287
6. CARVALHO, A. N. - CHOLEWA, J. W., Continuation and asymptotics of solutions to semilinear parabolic equations with critical nonlinearities, J. Math. Anal. Appl., 310 (2005), 557-578. Zbl1077.35031MR2022944DOI10.1016/j.jmaa.2005.02.024
7. CHEN, S. - TRIGGIANI, R., Proof of two conjectures on structural damping for elastic systems: The case a=1/2, Lecture Notes in Mathematics1354, Springer, 1988, 234-256. MR996678DOI10.1007/BFb0089601
8. CHOLEWA, J. W. - DLOTKO, T., Global Attractors in Abstract Parabolic Problems, Cambridge University Press, Cambridge, 2000. Zbl0954.35002MR1778284DOI10.1017/CBO9780511526404
9. CONTI, M. - PATA, V. - SQUASSINA, M., Singular limit of differential systems with memory, Indiana Univ. Math. J., 55 (2006), 169-215. Zbl1100.35018MR2207550DOI10.1512/iumj.2006.55.2661
10. DUNG, L. - NICOLAENKO, B., Exponential attractors in Banach spaces, J. Dynam. Differential Equations, 13 (2001), 791-806. Zbl1040.37069MR1860286DOI10.1023/A:1016676027666
11. EDEN, A. - FOIAS, C. - NICOLAENKO, B. - TEMAM, R., Exponential Attractors for Dissipative Evolution Equations, John Wiley and Sons, Ltd., Chichester, 1994. Zbl0842.58056MR1335230
12. EFENDIEV, M. - MIRANVILLE, A. - ZELIK, S., Exponential attractors for a nonlinear reaction-diffusion system in ${ℝ}^3$ , C. R. Acad. Sci. Paris Sr. I Math., 330 (2000), 713- 718. Zbl1151.35315MR1763916DOI10.1016/S0764-4442(00)00259-7
13. FABRIE, P. - GALUSINSKI, C. - MIRANVILLE, A. - ZELIK, S., Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Cont. Dyn. Systems, 10 (2004), 211-238. Zbl1060.35011MR2026192DOI10.3934/dcds.2004.10.211
14. GATTI, S. - GRASSELLI, M. - MIRANVILLE, A. - PATA, V., A construction of a robust family of exponential attractors, Proc. Amer. Math. Soc., 134 (2006), 117-127. Zbl1078.37047MR2170551DOI10.1090/S0002-9939-05-08340-1
15. GATTI, S. - GRASSELLI, M. - PATA, V., Exponential attractors for a phase-field model with memory and quadratic nonlinearities, Indiana Univ. Math. J., 53 (2004), 719- 754. Zbl1070.37056MR2086698DOI10.1512/iumj.2004.53.2413
16. HALE, J. K., Asymptotic Behavior of Dissipative Systems, AMS, Providence, R.I., 1988. MR941371
17. HENRY, D., Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 1981. Zbl0456.35001MR610244
18. LADYŽENSKAYA, O. A., Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991. 
19. LADYŽENSKAYA, O. A. - SOLONNIKOV, V. A. - URAL'CEVA, N. N., Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, AMS, Providence, R.I., 1967. MR241822
20. LI, DE-SHENG - ZHONG, CHEN-KUI, Global attractor for the Cahn-Hilliard system with fast growing nonlinearity, J. Diff. Equations, 149 (1998), 191-210. Zbl0912.35029MR1646238DOI10.1006/jdeq.1998.3429
21. MÁLEK, J. - PRAŽAK, D., Large time behavior via the method of $\mathrm{\ell }$-trajectories, J. Diff. Equations, 181 (2002), 243-279. MR1907143DOI10.1006/jdeq.2001.4087
22. MOLA, G., Global attractors for a three-dimensional conserved phase-field system with memory, Commun. Pure Appl. Anal., to appear. Zbl1144.35354MR2373219DOI10.3934/cpaa.2008.7.317
23. MOLA, G., Stability of global and exponential attractors for a three-dimensional conserved phase-field system with memory, submitted. Zbl1185.35027MR2373219DOI10.3934/cpaa.2008.7.317
24. PATA, V. - SQUASSINA, M., On the strongly damped wave equation, Comm. Math. Phys., 253 (2005), 511-533. Zbl1068.35077MR2116726DOI10.1007/s00220-004-1233-1
25. PATA, V. - ZELIK, S., Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 1495-1506. Zbl1113.35023MR2229785DOI10.1088/0951-7715/19/7/001
26. PATA, V. - ZELIK, S., A result on the existence of global attractors for semigroups of closed operators, Commun. Pure Appl. Anal., 6 (2007), 481-486. Zbl1152.47046MR2289833DOI10.3934/cpaa.2007.6.481
27. PATA, V. - ZUCCHI, A., Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529. Zbl0999.35014MR1907454
28. POLÁČIK, P., Parabolic equations: asymptotic behavior and dynamics on invariant manifolds, in: Handbook of Dynamical Systems Vol. 2, North-Holland, Amsterdam, 2002, 835-883. 
29. TEMAM, R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1988. Zbl0662.35001MR953967DOI10.1007/978-1-4684-0313-8
30. TRIEBEL, H., Interpolation Theory, Function Spaces, Differential Operators, Veb Deutscher, Berlin, 1978. MR500580
31. VON WAHL, W., Global solutions to evolution equations of parabolic type, in: Differential Equations in Banach Spaces, Proceedings, 1985, Springer-Verlag, Berlin, 1986, 254-266. MR872532DOI10.1007/BFb0099198
32. WEBB, G. F., Existence and asymptotic behavior for a strongly damped nonlinear wave equation, Can. J. Math., 32 (1980), 631-643. Zbl0414.35046MR586981DOI10.4153/CJM-1980-049-5