Exponential Attractors with Explicit Fractal Dimensions for Functional Differential Equations in Banach Spaces

• Wenjie Hu ^[1] ; Tomás Caraballo ^[2]
  1. [1] Hunan Normal University

     Hunan Normal University

     China

     
  2. [2] Universidad de Sevilla

     Universidad de Sevilla

     Sevilla, España

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 3, 2025
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • The aim of this paper is to propose a new method to construct exponential attractors for infinite dimensional dynamical systems in Banach spaces with explicit fractal dimensions. The approach is established by combining the squeezing properties and the covering of finite subspace of Banach spaces, which generalizes the method established in Hilbert spaces. The method is especially effective for functional differential equations in Banach spaces for which state decomposition of the linear part can be adopted to prove squeezing property. The theoretical results are applied to retarded functional differential equations and retarded reaction-diffusion equations for which the constructed exponential attractors possess explicit fractal dimensions that do not depend on the entropy number but only depend on the spectrum of the linear parts and Lipschitz constants of the nonlinear parts.

• Referencias bibliográficas
  • 1. Babin, A., Nicolaenko, B.: Exponential attractors of reaction-diffusion systems in an unbounded domain. J. Dyn. Differ. Equ. 7, 567–590...
  • 2. Caraballo, T., Crauel, H., Langa, J.A., Robinson, J.C.: The effect of noise on the Chafee–Infante equation: a nonlinear case study. Proc....
  • 3. Carvalho, A.N., Sonner, S.: Exponential attractors for semigroups in Banach spaces: theoretical results. Commun. Pure Appl. Anal. 12, 3047–3071...
  • 4. Chow, S.N., Mallet-Paret, J.: Integral averaging and bifurcation. J. Differ. Equ. 26, 112–159 (1977)
  • 5. Cui, H., Carvalho, A.N., Cunha, A.C., et al.: Smoothing and finite-dimensionality of uniform attractors in Banach spaces. J. Differ. Equ....
  • 6. Czaja, R., Efendiev, M.: Exponential attractors for nonautonomous equations Part I: Semilinear parabolic problems. J. Math. Anal. Appl....
  • 7. Eden, A., Foias, C., Nicolaenko, B., et al.: Exponential Attractors for Dissipative Evolution Equations. Research in Applied Mathematics,...
  • 8. Efendiev, M., Miranville, A., Zelik, S.: Exponential attractors for a nonlinear reaction-diffusion system in R3. C. R. Acad. Sci. 330,...
  • 9. Efendiev, M., Zelik, S., Miranville, A.: Exponential attractors and nite-dimensional reduction for non-autonomous dynamical systems. Proc....
  • 10. Efendiev, M., Yamamoto, Y., Yagi, A.: Exponential attractors for non-autonomous dissipative system. J. Math. Soc. Jpn 63, 647–673 (2011)
  • 11. Efendiev, M., Zelik, S.: Finite and infinite-dimensional exponential attractors for porous media equations. Proc. Lond. Math. Soc. 96,...
  • 12. Fabrie, P., Galusinski, C., Miranville, A., et al.: Uniform exponential attractors for a singularly perturbed damped wave equation. Discrete...
  • 13. Gatti, S., Miranville, A., Pata, V., et al.: Continuous families of exponential attractors for singularly perturbed equations with memory....
  • 14. Hale J.K., Lunel S.M.V.: Introduction to Functional Differential Equations. Springer Science Business Media (1993)
  • 15. Hu, W., Caraballo, T.: Hausdorff and fractal dimensions of attractors for functional differential equations in Banach spaces. J. Differ....
  • 16. Hu, W., Caraballo, T.: Pullback exponential attractors with explicit fractal dimensions for nonautonomous partial functional differential...
  • 17. Hu, W., Caraballo, T.: Exponential attractors for a nonlocal delayed reaction-diffusion equation on an unbounded domain. Proc. Am. Math....
  • 18. Mané, R.: On the Dimension of the Compact Invariant Sets of Certain Nonlinear Maps. Lecture Notes in Math., vol. 898, pp. 230–242. Springer,...
  • 19. Hammami, M., Mchiri, L., Netchaoui, S., Sonner, S.: Pullback exponential attractors for differential equations with variable delays. Disc....
  • 20. Netchaoui, S., Hammami, M., Caraballo, T.: Pullback exponential attractors for differential equations with delay. Disc. Cont. Dyn. Syst....
  • 21. Shirikyan, A., Zelik, S.: Exponential attractors for random dynamical systems and applications. Stoch. Partial Differ. Equ. Anal. Comput....
  • 22. Wang, R., Guo, B., Liu, W., et al.: Fractal dimension of random invariant sets and regular random attractors for stochastic hydrodynamical...
  • 23. Wu, J.: Theory and Applications of Partial Functional-Differential Equations. Springer, NewYork (1996)
  • 24. You, H., Yuan, R.: Global attractor for some partial functional differential equations with finite delay. Nonlinear Anal. TMA 72, 3566–3574...
  • 25. Zhao, C., Sun, W.: Global well-posedness and pullback attractors for a two-dimensional nonautonomous micropolar fluid flows with infinite...
  • 26. Zhou, S.: Random exponential attractors for stochastic reaction-diffusion equation with multiplicative noise in R3. J. Differ. Equ. 263,...
  • 27. Wang, Z., Zhou, S.: Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear...