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Dissipation and Compact Attractors

  • Jack K. Hale
Article

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.

Keywords

Unstable Manifold Global Attractor Center Manifold Discrete Dynamical System Continuous Dynamical System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Andronov A., Pontrjagin L.S. (1937). Systèmes grossiers. Dokl. Akad. Nauk SSSR 14, 247–251MATHGoogle Scholar
  2. 2.
    Angenent S. (1986). The Morse–Smale property for a semilinear parabolic equation. J. Diff. Eq. 62, 427–442MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Arrieta J., Carvalho A.N., Hale, J.K. (1992). A damped hyperbolic equation with a critical exponent. Comm. PDE 17, 841–866MATHMathSciNetGoogle Scholar
  4. 4.
    Babin A.V., Vishik M.I. (1992). Attractors of Evolution Equations. North-Holland, AmsterdamMATHGoogle Scholar
  5. 5.
    Ball J.M. (1997). Continuity properties of global attractors of generalized semiflows and the Navier-Stokes equations. J. Nonlinear Sci. 7, 475–502MATHMathSciNetCrossRefGoogle Scholar
  6. 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): 1024–1065MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Billotti J.E., LaSalle J.P. (1971). Periodic dissipative processes. Bull. Amer. Math. Soc. (N.S.) 6, 1082–1089MathSciNetGoogle Scholar
  8. 8.
    Browder F.E. (1959). On a generalization of the Schauder fixed point theorem. Duke Math. J. 26, 291–303MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Brunovsky P., Polačik P. (1997). The Morse-Smale structure of a generic reaction diffusion equation in higher space dimension. J. Diff. Eq. 135, 129–181MATHCrossRefGoogle Scholar
  10. 10.
    Chafee N., Infante E.F. (1974). A bifurcation problem for a nolinear parabolic equation. J. Appl. Anal. 4, 17–37MATHMathSciNetGoogle Scholar
  11. 11.
    Cholewa J.W., Hale J.K. (2000). Some counterexamples in dissipative systems. Dynamics of Continuous. Discrete Impulsive Syst. 7, 159–176MATHMathSciNetGoogle Scholar
  12. 12.
    Cruz M.A., Hale J.K. (1970). Stability of functional differential equations of neutral type. J. Diff. Eq. 7, 334–355MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Cooperman G. (1978). α-condensing maps and dissipative processes. Ph.D. Thesis, Brown University, Providence RI.Google Scholar
  14. 14.
    Dafermos C. (1978). Asymptotic behavior of solutions of evolutionary equations. In Nonlinear Evolution Equations M.G. Crandall (ed.), 103–123.Google Scholar
  15. 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: 759–776MATHGoogle Scholar
  16. 16.
    Faria T., Magalhães L. (1995b). Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity. J. Diff. Eq. 122, 201–224CrossRefMATHGoogle Scholar
  17. 17.
    Faria T., Magalhães L. (1995c). Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcations. J. Diff. Eq. 122, 181–200MATHCrossRefGoogle Scholar
  18. 18.
    Fiedler B., Rocha C. (1996). Heteroclinic orbits of semilinear parabolic equations. J. Diff. Eq. 156, 239–281MathSciNetCrossRefGoogle Scholar
  19. 19.
    Fiedler B., Rocha C. (2000). Orbit equivalence of global attractors of semilinear parabolic equations. Trans. Am. Math. Soc. 352, 257–284MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Fusco G., Rocha C. (1991). A permutation related to the dynamics of a scalar parabolic PDE. J. Diff. Eq. 91, 111–137MATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Gerstein V.M. (1970). On the theory of dissipative differential equations in a Banach space. Funk. Anal. I Prilzen 4, 99–100MathSciNetCrossRefGoogle Scholar
  22. 22.
    Gerstein V.M., Krasnoselskii M.A. (1968). Structure of the set of solutions of dissipative equations. Dokl. Akad. Nauk. SSSR 183, 267–269MathSciNetGoogle Scholar
  23. 23.
    Gobbino M., Sardella M. (1997). On the connectedness of attractors for dynamical systems. J. Diff. Eq. 133, 1–14MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Hale J.K. (1965). Sufficient conditions for stability and instability of autonomous functional differential equations. J. Diff. Eq. 1, 452–482MATHMathSciNetCrossRefGoogle Scholar
  25. 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.Google Scholar
  26. 26.
    Hale J.K. (1985a). Flows on center manifolds for scalar functional differential equations. Proc. Royal Soc. Edinburgh 101A: 193–201MathSciNetGoogle Scholar
  27. 27.
    Hale J.K. (1988). Asymptotic Behavior of Dissipative Systems. American Mathematical Society.Google Scholar
  28. 28.
    Hale J.K., LaSalle J.P., Slemrod M. (1972). Theory of a general class of dissipative processes. J. Math. Ana. Appl. 39, 171–191MathSciNetGoogle Scholar
  29. 29.
    Hale J.K., Lopes O. (1973). Fixed point theorems and dissipative processes. J. Diff. Eq. 13, 391–402MATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Hale, J.K., Magalhães, L., Oliva, W.M. (2002).Dynamics in Infinite Dimensions. Appl. Math. Sci. Vol. 47. Second edition. Springer-Verlag, BerlinGoogle Scholar
  31. 31.
    Hale J.K., Meyer K.R. (1967). A class of functional equations of neutral type. Mem. Amer. Math. Soc. 76, 1–65MathSciNetGoogle Scholar
  32. 32.
    Hale J.K., Raugel G. (1992). Convergence in gradient like systems. ZAMP 43, 63–124MATHMathSciNetCrossRefGoogle Scholar
  33. 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.Google Scholar
  34. 34.
    Hale J.K., Raugel G. (2003). Regularity, determining modes and Galerkin methods. J. Math. Pures. Appl. 82(9): 1075–1136MATHMathSciNetGoogle Scholar
  35. 35.
    Hale, J.K., Raugel, G. (2006). Infinite Dimensional Dynamical Systems. In preparation.Google Scholar
  36. 36.
    Hale J.K., Scheurle J. (1985). Smoothness of bounded solutions of nonlinear evolutionary equations. J. Diff. Eq. 56, 142–163MATHMathSciNetCrossRefGoogle Scholar
  37. 37.
    Hale J.K., Verduyn-Lunel, S. (1993). Introduction to functional differential equations. Appl. Math. Sci. Vol. 99. Springer-Verlag, BerlinGoogle Scholar
  38. 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.Google Scholar
  39. 39.
    Henry D. (1981). Geometric Theory of Semilinear Parabolic Equations. Springer, BerlinMATHGoogle Scholar
  40. 40.
    Henry D. (1985). Some infinite dimensional Morse–Smale systems defined by parabolic differential equations. J. Diff. Eq. 59, 165–205MATHMathSciNetCrossRefGoogle Scholar
  41. 41.
    Henry D. (1987). Topics in analysis. Pub. Mat. UAB 31, 29–84MATHMathSciNetGoogle Scholar
  42. 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.Google Scholar
  43. 43.
    Hirsch M.W. (1988). Stability convergence in strongly monotone dynamical systems. J. Reine Angew. Math. 383, 1–53MATHMathSciNetGoogle Scholar
  44. 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, 193–218MATHMathSciNetGoogle Scholar
  45. 45.
    Jones G.S., Yorke J. (1969). The existence and nonexistence of critical points in bounded flows. J. Diff. Eq. 6, 238–247MATHMathSciNetCrossRefGoogle Scholar
  46. 46.
    LaSalle, J.P. (1976). The stability of dynamical systems. CBMS Regional Conf. Ser. SIAM 25.Google Scholar
  47. 47.
    Ladyzenskaya O.A. (1987). On the determination of minimal global attractors for the Stokes equation and other differential operators. Russian Math. Surveys 42, 27–73CrossRefGoogle Scholar
  48. 48.
    Levin J.J., Nohel J.A. (1964). On a nonlinear delay equation. J. Math. Anal. Appl. 8, 31–44MATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    Levinson N. (1944). Transformation theory of nonlinear differential equations of the second order. Ann. Math. 45(2): 724–737MathSciNetGoogle Scholar
  50. 50.
    Mallet-Paret J. (1977). Generic periodic solutions of functional differential equations. J. Diff. Eq. 25, 163–183MATHMathSciNetCrossRefGoogle Scholar
  51. 51.
    Massatt P. (1980). Some properties of α-condensing maps. Ann. Mat. Pura Appl. 125(4): 101–115MATHMathSciNetCrossRefGoogle Scholar
  52. 52.
    Massatt P. (1983) Attractivity properties of α-contractions. J. Diff. Eq. 48, 326–333MATHMathSciNetCrossRefGoogle Scholar
  53. 53.
    Massera J.L. (1950). The existence of periodic solutions ofsystems of differential equations. Duke Math. J. 17, 457–475MATHMathSciNetCrossRefGoogle Scholar
  54. 54.
    Matano H. (1978). Convergence of solutions of one-dimensional semilinear parabolic equations. J. Math. Kyoto Univ. 18, 224–243MathSciNetGoogle Scholar
  55. 55.
    Nussbaum R. (1972). Some asymptotic fixed point theorems. Trans. Am. Math. Soc. 171, 349–375MATHMathSciNetCrossRefGoogle Scholar
  56. 56.
    Oliva W.M. (1969). Functional differential equations on compact manifolds and an approximation theorem. J. Diff. Eq. 5, 483–496MATHMathSciNetCrossRefGoogle Scholar
  57. 57.
    Oliva, W.M. (1982) Stability of Morse-Smale maps. D.Mat. Applicada-IME-Un. Saõ Paulo-Brasil 1–49Google Scholar
  58. 58.
    Palis J., de Melo W. (1982). Geometric Theory of Dynamical Systems. Springer-Verlag, New YorkMATHGoogle Scholar
  59. 59.
    Peixoto M.M. (1959). On structural stability. Ann. Math. 69, 189–222MathSciNetCrossRefGoogle Scholar
  60. 60.
    Pliss V. (1966). Nonlocal Problems in the Theory of Oscillations. Academic Press, New YorkGoogle Scholar
  61. 61.
    Polačik P. (1991). Complicated dynamics in scalar semilinear parabolic equations in higher space dimensions.J. Diff. Eq. 89, 244–271CrossRefMATHGoogle Scholar
  62. 62.
    Polačik P. (1999). Persistent stable connections in a class of reaction-diffusion equations. J. Diff. Eq. 156, 182–210CrossRefMATHGoogle Scholar
  63. 63.
    Polačik, P. (2002). Parabolic equations: asymptotic behavior and dynamics on invariant manifolds. Handbook of Dynamical Systems. North-Holland, Amsterdam, Vol. 2, pp. 835–883Google Scholar
  64. 64.
    Polačik P., Rybakowski K. (1995). Embedding vector fields into Dirichlet BVP’s. Ann. Scuola Norm. Sup. Pisa 21, 737–749Google Scholar
  65. 65.
    Polačik P., Rybakowski K. (1996). Nonconvergent bounded trajectories in semilinear heat equations. J. Diff. Eq. 124, 472–494CrossRefMATHGoogle Scholar
  66. 66.
    Prizzi M. (1998). Realizing vector fields without loss of derivatives. Ann. Scuola Norm. Sup. Pisa 27, 289–307MATHMathSciNetGoogle Scholar
  67. 67.
    Raugel, G. (1995). Dynamics of partial differential equations on thin domains. In Dynamical Systems, R. Johnson (ed.) LNM 1609, Springer, Berlin, pp. 208–315Google Scholar
  68. 68.
    Raugel, G. (2002). Global attractors in partial differential equations. In Handbook of Dynamical Systems. North-Holland, Amsterdam, Vol. 2, pp. 885–982Google Scholar
  69. 69.
    Ruiz A. (1992). Unique continuation for weak solutions of the wave equation plus a potential. J. Math. Pures Appl. 71, 455–467MATHMathSciNetGoogle Scholar
  70. 70.
    Rybakowski K.P. (1994). Realization of arbitrary vector fields on invariant manifolds of delay equations. J. Diff. Eq. 114, 222–231MATHMathSciNetCrossRefGoogle Scholar
  71. 71.
    Sell G.R., You Y. (2002). Dynamics of Evolutionary Equations. Springer-Verlag, New YorkMATHGoogle Scholar
  72. 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, 428–438MathSciNetGoogle Scholar
  73. 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, 91–102MathSciNetCrossRefGoogle Scholar
  74. 74.
    Simon L. (1983). Asymptotics for a class of nonlinear evolution equations. Ann. of Math. 118(2): 525–571MATHMathSciNetCrossRefGoogle Scholar
  75. 75.
    Smale S. (1967). Differentiable dynamic systems. Bull. Amer. Math. Soc. 73, 747–817MATHMathSciNetCrossRefGoogle Scholar
  76. 76.
    Temam R. (1997). Infinite Dimensional Dynamical Systems in Mechanics and Physics, Second Edition. Springer-Verlag, BerlinGoogle Scholar
  77. 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, 17–22Google Scholar

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© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

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