Skip to main content
SpringerLink
Log in

Dissipation and Compact Attractors

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

We present an approach to the study of the qualitative theory of infinite dimensional dynamical systems. In finite dimensions, most of the success has been with the discussion of dynamics on sets which are invariant and compact. In the infinite dimensional case, the appropriate setting is to consider the dynamics on the maximal compact invariant set. In dissipative systems, this corresponds to the compact global attractor. Most of the time is devoted to necessary and sufficient conditons for the existence of the compact global attractor. Several important applications are given as well as important results on the qualitative properties of the flow on the attractor.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Andronov A., Pontrjagin L.S. (1937). Systèmes grossiers. Dokl. Akad. Nauk SSSR 14, 247–251

    MATH  Google Scholar 

  2. Angenent S. (1986). The Morse–Smale property for a semilinear parabolic equation. J. Diff. Eq. 62, 427–442

    Article  MATH  MathSciNet  Google Scholar 

  3. Arrieta J., Carvalho A.N., Hale, J.K. (1992). A damped hyperbolic equation with a critical exponent. Comm. PDE 17, 841–866

    MATH  MathSciNet  Google Scholar 

  4. Babin A.V., Vishik M.I. (1992). Attractors of Evolution Equations. North-Holland, Amsterdam

    MATH  Google Scholar 

  5. Ball J.M. (1997). Continuity properties of global attractors of generalized semiflows and the Navier-Stokes equations. J. Nonlinear Sci. 7, 475–502

    Article  MATH  MathSciNet  Google Scholar 

  6. Bardos C., Lebeau G., Rauch J. (1992). Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optimization 30(5): 1024–1065

    Article  MATH  MathSciNet  Google Scholar 

  7. Billotti J.E., LaSalle J.P. (1971). Periodic dissipative processes. Bull. Amer. Math. Soc. (N.S.) 6, 1082–1089

    MathSciNet  Google Scholar 

  8. Browder F.E. (1959). On a generalization of the Schauder fixed point theorem. Duke Math. J. 26, 291–303

    Article  MATH  MathSciNet  Google Scholar 

  9. Brunovsky P., Polačik P. (1997). The Morse-Smale structure of a generic reaction diffusion equation in higher space dimension. J. Diff. Eq. 135, 129–181

    Article  MATH  Google Scholar 

  10. Chafee N., Infante E.F. (1974). A bifurcation problem for a nolinear parabolic equation. J. Appl. Anal. 4, 17–37

    MATH  MathSciNet  Google Scholar 

  11. Cholewa J.W., Hale J.K. (2000). Some counterexamples in dissipative systems. Dynamics of Continuous. Discrete Impulsive Syst. 7, 159–176

    MATH  MathSciNet  Google Scholar 

  12. Cruz M.A., Hale J.K. (1970). Stability of functional differential equations of neutral type. J. Diff. Eq. 7, 334–355

    Article  MATH  MathSciNet  Google Scholar 

  13. Cooperman G. (1978). α-condensing maps and dissipative processes. Ph.D. Thesis, Brown University, Providence RI.

  14. Dafermos C. (1978). Asymptotic behavior of solutions of evolutionary equations. In Nonlinear Evolution Equations M.G. Crandall (ed.), 103–123.

  15. Faria T., Magalhães L. (1995a). Realization of ordinary differential equations by retarded functional differential equations in neighborhoods of equilibrium points. Proc. Roy. Soc. Edinburgh 125A: 759–776

    MATH  Google Scholar 

  16. Faria T., Magalhães L. (1995b). Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity. J. Diff. Eq. 122, 201–224

    Article  MATH  Google Scholar 

  17. Faria T., Magalhães L. (1995c). Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcations. J. Diff. Eq. 122, 181–200

    Article  MATH  Google Scholar 

  18. Fiedler B., Rocha C. (1996). Heteroclinic orbits of semilinear parabolic equations. J. Diff. Eq. 156, 239–281

    Article  MathSciNet  Google Scholar 

  19. Fiedler B., Rocha C. (2000). Orbit equivalence of global attractors of semilinear parabolic equations. Trans. Am. Math. Soc. 352, 257–284

    Article  MATH  MathSciNet  Google Scholar 

  20. Fusco G., Rocha C. (1991). A permutation related to the dynamics of a scalar parabolic PDE. J. Diff. Eq. 91, 111–137

    Article  MATH  MathSciNet  Google Scholar 

  21. Gerstein V.M. (1970). On the theory of dissipative differential equations in a Banach space. Funk. Anal. I Prilzen 4, 99–100

    Article  MathSciNet  Google Scholar 

  22. Gerstein V.M., Krasnoselskii M.A. (1968). Structure of the set of solutions of dissipative equations. Dokl. Akad. Nauk. SSSR 183, 267–269

    MathSciNet  Google Scholar 

  23. Gobbino M., Sardella M. (1997). On the connectedness of attractors for dynamical systems. J. Diff. Eq. 133, 1–14

    Article  MATH  MathSciNet  Google Scholar 

  24. Hale J.K. (1965). Sufficient conditions for stability and instability of autonomous functional differential equations. J. Diff. Eq. 1, 452–482

    Article  MATH  MathSciNet  Google Scholar 

  25. Hale, J.K. (1985). Asymptotic behavior and dynamics in infinite dimensions. In Res. Notes in Math., Hale and Martinez-Amores (eds.), Pitman, London, Vol. 132, pp. 1–41.

  26. Hale J.K. (1985a). Flows on center manifolds for scalar functional differential equations. Proc. Royal Soc. Edinburgh 101A: 193–201

    MathSciNet  Google Scholar 

  27. Hale J.K. (1988). Asymptotic Behavior of Dissipative Systems. American Mathematical Society.

  28. Hale J.K., LaSalle J.P., Slemrod M. (1972). Theory of a general class of dissipative processes. J. Math. Ana. Appl. 39, 171–191

    MathSciNet  Google Scholar 

  29. Hale J.K., Lopes O. (1973). Fixed point theorems and dissipative processes. J. Diff. Eq. 13, 391–402

    Article  MATH  MathSciNet  Google Scholar 

  30. Hale, J.K., Magalhães, L., Oliva, W.M. (2002).Dynamics in Infinite Dimensions. Appl. Math. Sci. Vol. 47. Second edition. Springer-Verlag, Berlin

  31. Hale J.K., Meyer K.R. (1967). A class of functional equations of neutral type. Mem. Amer. Math. Soc. 76, 1–65

    MathSciNet  Google Scholar 

  32. Hale J.K., Raugel G. (1992). Convergence in gradient like systems. ZAMP 43, 63–124

    Article  MATH  MathSciNet  Google Scholar 

  33. Hale, J.K., Raugel, G. (1993). Attractors for dissipative evolutionary equations. In Equadiff 91, International Conference on Differential Equations, Barcelona (1991), World Scientific, Singapore, pp. 3–22.

  34. Hale J.K., Raugel G. (2003). Regularity, determining modes and Galerkin methods. J. Math. Pures. Appl. 82(9): 1075–1136

    MATH  MathSciNet  Google Scholar 

  35. Hale, J.K., Raugel, G. (2006). Infinite Dimensional Dynamical Systems. In preparation.

  36. Hale J.K., Scheurle J. (1985). Smoothness of bounded solutions of nonlinear evolutionary equations. J. Diff. Eq. 56, 142–163

    Article  MATH  MathSciNet  Google Scholar 

  37. Hale J.K., Verduyn-Lunel, S. (1993). Introduction to functional differential equations. Appl. Math. Sci. Vol. 99. Springer-Verlag, Berlin

  38. Haraux, A. (1985). Two remarks on hyperbolic dissipative problems. Nonlinear Partial Differential Equations and their Applications. Coll‘ege de France Seminar, Vol. VII (Paris 1983-1985), Research Notes in Math. Pitman, London, Vol 122, pp. 161–179.

  39. Henry D. (1981). Geometric Theory of Semilinear Parabolic Equations. Springer, Berlin

    MATH  Google Scholar 

  40. Henry D. (1985). Some infinite dimensional Morse–Smale systems defined by parabolic differential equations. J. Diff. Eq. 59, 165–205

    Article  MATH  MathSciNet  Google Scholar 

  41. Henry D. (1987). Topics in analysis. Pub. Mat. UAB 31, 29–84

    MATH  MathSciNet  Google Scholar 

  42. Henry, D. (2005). Perturbation of the Boundary in Boundary Value Problems in Partial Differential Equations. London Math. Soc. Lect. Notes Series Vol. 318. Cambridge University, Press, Cambridge.

  43. Hirsch M.W. (1988). Stability convergence in strongly monotone dynamical systems. J. Reine Angew. Math. 383, 1–53

    MATH  MathSciNet  Google Scholar 

  44. Iwasaki N. (1969). Local decay of solutions for symmetric hyperbolic systems and coercive boundary conditions in exterior domains. Pub. Res. Inst. Math. Sci. Kyoto Univ. 5, 193–218

    MATH  MathSciNet  Google Scholar 

  45. Jones G.S., Yorke J. (1969). The existence and nonexistence of critical points in bounded flows. J. Diff. Eq. 6, 238–247

    Article  MATH  MathSciNet  Google Scholar 

  46. LaSalle, J.P. (1976). The stability of dynamical systems. CBMS Regional Conf. Ser. SIAM 25.

  47. Ladyzenskaya O.A. (1987). On the determination of minimal global attractors for the Stokes equation and other differential operators. Russian Math. Surveys 42, 27–73

    Article  Google Scholar 

  48. Levin J.J., Nohel J.A. (1964). On a nonlinear delay equation. J. Math. Anal. Appl. 8, 31–44

    Article  MATH  MathSciNet  Google Scholar 

  49. Levinson N. (1944). Transformation theory of nonlinear differential equations of the second order. Ann. Math. 45(2): 724–737

    MathSciNet  Google Scholar 

  50. Mallet-Paret J. (1977). Generic periodic solutions of functional differential equations. J. Diff. Eq. 25, 163–183

    Article  MATH  MathSciNet  Google Scholar 

  51. Massatt P. (1980). Some properties of α-condensing maps. Ann. Mat. Pura Appl. 125(4): 101–115

    Article  MATH  MathSciNet  Google Scholar 

  52. Massatt P. (1983) Attractivity properties of α-contractions. J. Diff. Eq. 48, 326–333

    Article  MATH  MathSciNet  Google Scholar 

  53. Massera J.L. (1950). The existence of periodic solutions ofsystems of differential equations. Duke Math. J. 17, 457–475

    Article  MATH  MathSciNet  Google Scholar 

  54. Matano H. (1978). Convergence of solutions of one-dimensional semilinear parabolic equations. J. Math. Kyoto Univ. 18, 224–243

    MathSciNet  Google Scholar 

  55. Nussbaum R. (1972). Some asymptotic fixed point theorems. Trans. Am. Math. Soc. 171, 349–375

    Article  MATH  MathSciNet  Google Scholar 

  56. Oliva W.M. (1969). Functional differential equations on compact manifolds and an approximation theorem. J. Diff. Eq. 5, 483–496

    Article  MATH  MathSciNet  Google Scholar 

  57. Oliva, W.M. (1982) Stability of Morse-Smale maps. D.Mat. Applicada-IME-Un. Saõ Paulo-Brasil 1–49

  58. Palis J., de Melo W. (1982). Geometric Theory of Dynamical Systems. Springer-Verlag, New York

    MATH  Google Scholar 

  59. Peixoto M.M. (1959). On structural stability. Ann. Math. 69, 189–222

    Article  MathSciNet  Google Scholar 

  60. Pliss V. (1966). Nonlocal Problems in the Theory of Oscillations. Academic Press, New York

    Google Scholar 

  61. Polačik P. (1991). Complicated dynamics in scalar semilinear parabolic equations in higher space dimensions.J. Diff. Eq. 89, 244–271

    Article  MATH  Google Scholar 

  62. Polačik P. (1999). Persistent stable connections in a class of reaction-diffusion equations. J. Diff. Eq. 156, 182–210

    Article  MATH  Google Scholar 

  63. Polačik, P. (2002). Parabolic equations: asymptotic behavior and dynamics on invariant manifolds. Handbook of Dynamical Systems. North-Holland, Amsterdam, Vol. 2, pp. 835–883

  64. Polačik P., Rybakowski K. (1995). Embedding vector fields into Dirichlet BVP’s. Ann. Scuola Norm. Sup. Pisa 21, 737–749

    Google Scholar 

  65. Polačik P., Rybakowski K. (1996). Nonconvergent bounded trajectories in semilinear heat equations. J. Diff. Eq. 124, 472–494

    Article  MATH  Google Scholar 

  66. Prizzi M. (1998). Realizing vector fields without loss of derivatives. Ann. Scuola Norm. Sup. Pisa 27, 289–307

    MATH  MathSciNet  Google Scholar 

  67. Raugel, G. (1995). Dynamics of partial differential equations on thin domains. In Dynamical Systems, R. Johnson (ed.) LNM 1609, Springer, Berlin, pp. 208–315

  68. Raugel, G. (2002). Global attractors in partial differential equations. In Handbook of Dynamical Systems. North-Holland, Amsterdam, Vol. 2, pp. 885–982

  69. Ruiz A. (1992). Unique continuation for weak solutions of the wave equation plus a potential. J. Math. Pures Appl. 71, 455–467

    MATH  MathSciNet  Google Scholar 

  70. Rybakowski K.P. (1994). Realization of arbitrary vector fields on invariant manifolds of delay equations. J. Diff. Eq. 114, 222–231

    Article  MATH  MathSciNet  Google Scholar 

  71. Sell G.R., You Y. (2002). Dynamics of Evolutionary Equations. Springer-Verlag, New York

    MATH  Google Scholar 

  72. Sil’nikov L.P. (1968). On the generation of periodic motion from trajectories doubly asymptotic to an equilibrium of saddle type. Math. Sb. 6, 428–438

    MathSciNet  Google Scholar 

  73. Sil’nikov L.P. (1970). A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of a saddle-focus type. Math. Sb. 10, 91–102

    Article  MathSciNet  Google Scholar 

  74. Simon L. (1983). Asymptotics for a class of nonlinear evolution equations. Ann. of Math. 118(2): 525–571

    Article  MATH  MathSciNet  Google Scholar 

  75. Smale S. (1967). Differentiable dynamic systems. Bull. Amer. Math. Soc. 73, 747–817

    Article  MATH  MathSciNet  Google Scholar 

  76. Temam R. (1997). Infinite Dimensional Dynamical Systems in Mechanics and Physics, Second Edition. Springer-Verlag, Berlin

    Google Scholar 

  77. Zelenyak T.J. (1968). Stabilization of solutions of boundary value problems for a second order parabolic problem with one space variable. Diff. Eq. 4, 17–22

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332-0160, USA

    Jack K. Hale

Authors
  1. Jack K. Hale
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jack K. Hale.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hale, J.K. Dissipation and Compact Attractors. J Dyn Diff Equat 18, 485–523 (2006). https://doi.org/10.1007/s10884-006-9021-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-006-9021-6

Keywords

Search

Navigation