ArticleOriginal scientific text

Title

Invariant manifolds for one-dimensional parabolic partial differential equations of second order

Authors 1

Affiliations

  1. Institute of Mathematics, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

Bibliography

  1. H. Amann, Existence and multiplicity theorems for semi-linear elliptic boundary value problems, Math. Z. 150 (1976), 281-295.
  2. H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), 620-709.
  3. H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, Differential Integral Equations (1990), 13-75.
  4. S. Angenent, The Morse-Smale property for a semi-linear parabolic equation, J. Differential Equations 62 (1986), 427-442.
  5. S. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math. 390 (1988), 79-96.
  6. S. Angenent, Nonlinear analytic semiflows, Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), 91-107.
  7. N. Aronszajn, Differentiability of Lipschitzian mappings between Banach spaces, Studia Math. 57 (1976), 147-190.
  8. P. Brunovský and B. Fiedler, Numbers of zeros on invariant manifolds in reaction-diffusion equations, Nonlinear Anal. 10 (1986), 179-193.
  9. P. Brunovský and B. Fiedler, Connecting orbits in scalar reaction diffusion equations. II: The complete solution, J. Differential Equations 81 (1989), 106-135.
  10. P. Brunovský, P. Poláčik and B. Sandstede, Convergence in general periodic parabolic equations in one space dimension, Nonlinear Anal. 18 (1992), 209-215.
  11. M. Chen, X.-Y. Chen and J. K. Hale, Structural stability for time-periodic one-dimensional parabolic equations, J. Differential Equations 96 (1992), 355-418.
  12. X.-Y. Chen and H. Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equation, ibid. 78 (1989), 160-190.
  13. X.-Y. Chen and P. Poláčik, Gradient-like structure and Morse decompositions for time-periodic one-dimensional parabolic equations, J. Dynam. Differential Equations 7 (1995), 73-107.
  14. S.-N. Chow and J. K. Hale, Methods of Bifurcation Theory, Grundlehren Math. Wiss. 251, Springer, New York, 1982.
  15. S.-N. Chow, X.-B. Lin and K. Lu, Smooth invariant foliations in infinite dimensional spaces, J. Differential Equations 94 (1991), 266-291.
  16. S.-N. Chow, K. Lu and J. Mallet-Paret, Floquet bundles for scalar parabolic equations, Arch. Rational Mech. Anal. 129 (1995), 245-304.
  17. K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985.
  18. C. Foiaş and J.-C. Saut, On the smoothness of the nonlinear spectral manifolds of Navier-Stokes equations, Indiana Univ. Math. J. 33 (1984), 911-926.
  19. G. Fusco and W. M. Oliva, Jacobi matrices and transversality, Proc. Roy. Soc. Edinburgh Sect. A 109 (1988), 231-243.
  20. J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys Monographs 25, Amer. Math. Soc., Providence, R.I., 1988.
  21. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer, Berlin, 1981.
  22. D. Henry, Some infinite dimensional Morse-Smale systems defined by parabolic partial differential equations, J. Differential Equations 53 (1985), 401-458.
  23. P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Res. Notes Math. Ser. 247, Longman Sci. Tech., Harlow, 1991.
  24. M. W. Hirsch, Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math. 383 (1988), 1-53.
  25. M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Math. 583, Springer, Berlin, 1977.
  26. H. Koch, Finite dimensional aspects of semilinear parabolic equations, J. Dynam. Differential Equations 8 (1996), 177-202.
  27. H.-H. Kuo, Gaussian Measures in Banach Spaces, Lecture Notes in Math. 463, Springer, Berlin, 1975.
  28. S. Lang, Differential Manifolds, Addison-Wesley, Reading, Mass., 1972.
  29. H. Matano, Convergence of solutions of one-dimensional semilinear parabolic equations, J. Math. Kyoto Univ. 18 (1978), 221-227.
  30. H. Matano, Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), 401-441.
  31. J. Mierczyński, On monotone trajectories, Proc. Amer. Math. Soc. 113 (1991), 537-544.
  32. J. Mierczyński, P-arcs in strongly monotone discrete-time dynamical systems, Differential Integral Equations 7 (1994), 1473-1494.
  33. M. Miklavčič, Stability for semilinear parabolic equations with noninvertible linear operator, Pacific J. Math. 118 (1985), 199-214.
  34. K. Nickel, Gestaltaussagen über Lösungen parabolischer Differentialgleichungen, J. Reine Angew. Math. 211 (1962), 78-94.
  35. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. 44, Springer, New York, 1983.
  36. R. R. Phelps, Gaussian null sets and differentiability of Lipschitz maps on Banach spaces, Pacific J. Math. 77 (1978), 523-531.
  37. P. Poláčik, Domains of attraction of equilibria and monotonicity properties of convergent trajectories in parabolic systems admitting strong comparison principle, J. Reine Angew. Math. 400 (1989), 32-56.
  38. W. Shen and Y. Yi, On minimal sets of scalar parabolic equations with skew-product structures, Trans. Amer. Math. Soc. 347 (1995), 4413-4431.
  39. A. V. Skorokhod [A. V. Skorohod], Integration in Hilbert Space, translated from the Russian by K. Wickwire, Ergeb. Math. Grenzgeb. 79, Springer, New York, 1974.
  40. J. Smillie, Competitive and cooperative tridiagonal systems of differential equations, SIAM J. Math. Anal. 15 (1984), 530-534.
  41. P. Takáč, Convergence to equilibrium on invariant d-hypersurfaces for strongly increasing discrete-time semigroups, J. Math. Anal. Appl. 148 (1990), 223-244.
  42. P. Takáč, Domains of attraction of generic οmega-limit sets for strongly monotone discrete-time semigroups, J. Reine Angew. Math. 423 (1992), 101-173.
  43. T. I. Zelenyak, Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable, Differentsialnye Uravneniya 4 (1968), 34-45; English transl.: Differential Equations 4 (1968), 17-22.
Pages:
285-314
Main language of publication
English
Received
1997-03-12
Accepted
1997-05-28
Published
1998
Exact and natural sciences