On dimension of attractors of reaction-diffusion equations with periodic right-hand side
Abstract
In this paper we study the finite-dimensionality of the global attractor of a discrete dynamical system generated by a reaction-diffusion equation with non-differentiable nonlinear term and periodic right-hand side. The existence of an exponential attractor is also proved. Explicit estimates of the fractal dimension are given.
References
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5. F. Balibrea, J. Valero, Estimates of dimension of attractors of reaction-diffusion equations in the nondifferentiable case, C. R. Acad. Sci. Paris 325 (1997), 759-764.
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13. A. Eden, C. Foias, B. Nicolaenko, R. Temam, Ensembles inertielles pour des équations d'évolution dissipatives, C. R. Acad. Sci. Paris, Série I 310 (1990), 559-562.
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15. A. Eden, C. Foias, B. Nicolaenko, R. Temam, Exponential attractors for dissipative evolutionary equations, Research in Applied Mathematics, No. 37, John Wiley & Sons, Masson, 1995.
16. A. Eden, B. Michaux, J.M. Rakotoson, Doubly nonlinear parabolic-type equations as dynamical systems, J. Dynam. Differential Equations 3 (1991), 87-131.
17. A. Eden, J.M. Rakotoson, Exponential attractors for a doubly nonlinear equation, J. Math. Anal. Appl. 185 (1994), 321-339.
18. P. Fabrie, C. Galusinski, Exponential attractors for a partially dissipative reaction system, Asymptotic Analysis 12 (1996), 329-354.
19. J.K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs 25, AMS, Providence, 1988.
20. O.A. Ladyzhenskaya, Some comments to my papers on the theory of attractors for abstract semigroups (in russian), Zap. Nauchn. Sem. LOMI 182 (1990), 102-112 (English translation in J. Soviet Math, 62 (1992), 1789-1794).
21. M. Marion, Attractors for reaction-diffusion equations: existence and estimate of their dimension, Appl. Anal. 25 (1987), 101-147.
22. M. Marion, Finite-dimensional attractors associated with partly dissipative reaction-diffusion systems, SIAM J. Math. Anal. 20 (1989), 816-844.
23. G. Metivier, Valeurs propres d'opérators dé finis par la restriction de systèmes variationnels a des sous-espaces, J. Math. Pures Appl. 57 (1978), 133-156.
24. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988.
25. Yi. Zhao, The global attractor of infinite-dimensional dynamical systems governed by a class of nonlinear parabolic variational inequalities and associated control problems, Appl. Anal. 54 (1994), 163-180.
2. A.V. Babin, M.I. Vishik, Attracteurs maximaux dans les équations aux dérivées partielles, Nonlinear Partial Differential Equations and Their Applications (Ed. H. Brezis and J.L. Lions) Vol. 7, Research Notes in Math No. 122, Pitman (1985), 11-34.
3. A.V. Babin, M.I. Vishik, Attractors of Evolution Equations, Nauka, Moscow 1989.
4. A.V. Babin, M.I. Vishik, Attractors of partial differential evolution equations in an unbounded domain, Proc. Roy. Soc. Edinburgh 116A (1990), 221-243.
5. F. Balibrea, J. Valero, Estimates of dimension of attractors of reaction-diffusion equations in the nondifferentiable case, C. R. Acad. Sci. Paris 325 (1997), 759-764.
6. F. Balibrea, J. Valero, On dimension of attractors of differential inclusions and reaction-diffusion equations, Preprint, Departamento de Matemáticas, Universidad de Murcia 1997.
7. F. Balibrea, J. Valero, On dimension of attractors of differential inclusions and reaction-diffusion equations, Discrete. Contin. Dynam. Systems 5 (1999), 515-528.
8. V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei, Bucuresti 1976.
9. H. Brezis, Problémes unilatéraux, J. Math. Pures Appl. 51 (1972), 1-168.
10. H. Brezis, Análisis Funcional, Alianza Editorial, Madrid, 1984.
11. V.V. Chepyzhov, M.I. Vishik, Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl. 73 (1994), 279-333.
12. A. Eden, C. Foias, V. Kalantarov, A remark on two constructions of exponential attractors for ?-contractions, J. Dynam. Differential Equations 10 (1998), 37-45.
13. A. Eden, C. Foias, B. Nicolaenko, R. Temam, Ensembles inertielles pour des équations d'évolution dissipatives, C. R. Acad. Sci. Paris, Série I 310 (1990), 559-562.
14. A. Eden, C. Foias, B. Nicolaenko, R. Temam, Inertial sets for dissipative evolution equations. Part I: construction and applications, IMA Preprint No. 812, 1991, 135.
15. A. Eden, C. Foias, B. Nicolaenko, R. Temam, Exponential attractors for dissipative evolutionary equations, Research in Applied Mathematics, No. 37, John Wiley & Sons, Masson, 1995.
16. A. Eden, B. Michaux, J.M. Rakotoson, Doubly nonlinear parabolic-type equations as dynamical systems, J. Dynam. Differential Equations 3 (1991), 87-131.
17. A. Eden, J.M. Rakotoson, Exponential attractors for a doubly nonlinear equation, J. Math. Anal. Appl. 185 (1994), 321-339.
18. P. Fabrie, C. Galusinski, Exponential attractors for a partially dissipative reaction system, Asymptotic Analysis 12 (1996), 329-354.
19. J.K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs 25, AMS, Providence, 1988.
20. O.A. Ladyzhenskaya, Some comments to my papers on the theory of attractors for abstract semigroups (in russian), Zap. Nauchn. Sem. LOMI 182 (1990), 102-112 (English translation in J. Soviet Math, 62 (1992), 1789-1794).
21. M. Marion, Attractors for reaction-diffusion equations: existence and estimate of their dimension, Appl. Anal. 25 (1987), 101-147.
22. M. Marion, Finite-dimensional attractors associated with partly dissipative reaction-diffusion systems, SIAM J. Math. Anal. 20 (1989), 816-844.
23. G. Metivier, Valeurs propres d'opérators dé finis par la restriction de systèmes variationnels a des sous-espaces, J. Math. Pures Appl. 57 (1978), 133-156.
24. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988.
25. Yi. Zhao, The global attractor of infinite-dimensional dynamical systems governed by a class of nonlinear parabolic variational inequalities and associated control problems, Appl. Anal. 54 (1994), 163-180.
BalibreaF., & ValeroJ. (1999). On dimension of attractors of reaction-diffusion equations with periodic right-hand side. Annales Mathematicae Silesianae, 13, 61-71. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14137
Francisco Balibrea
Departamento de Matemáticas, Universidad de Murcia, Spain Spain
Departamento de Matemáticas, Universidad de Murcia, Spain Spain
José Valero
CEU San Pablo Elche, Spain Spain
CEU San Pablo Elche, Spain Spain
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