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Singular solutions of a nonlinear elliptic equation and an infinite dimensional dynamical system

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Functional-Analytic Methods for Partial Differential Equations

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References

  1. C.J. Amick and K. Kirchgässner: Solitary water waves in the presence of surface tension, Arch. Rat. Mech. Anal., to appear.

    Google Scholar 

  2. C.J. Amick and R.E.L. Turner: Small internal waves in two-fluid systems, Univ. Wisc. Tech. Summary Report No 89-4.

    Google Scholar 

  3. S.B. Angenent: The Morse-Smale property for a semi-linear parabolic equation, J. Differential Equations, 62 (1986), 427–442.

    Article  MathSciNet  MATH  Google Scholar 

  4. S.B. Angenent and B. Fiedler: The dynamics of rotating waves in scalar reaction diffusion equations, Trans. Amer. Math. Soc., 307 (1988), 545–568.

    Article  MathSciNet  MATH  Google Scholar 

  5. P. Aviles: On isolated singularities in some nonlinear partial differential equations, Indiana Univ. Math. J., 32 (1983), 773–790.

    Article  MathSciNet  MATH  Google Scholar 

  6. H. Brezis and E.H. Lieb: Long range atomic potentials in Thomas-Fermi theory, Comm. Math. Phys., 65 (1979), 231–246.

    Article  MathSciNet  MATH  Google Scholar 

  7. K.-C. Chang: Infinite dimensional Morse theory and its applications, Seminaire de Math. Superieures, 97, Presses de l'Université de Montreal, Quebec, 1985.

    Google Scholar 

  8. X.-Y. Chen and H. Matano: Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations, J. Differential Equations, 78 (1989), 160–190.

    Article  MathSciNet  MATH  Google Scholar 

  9. X-Y. Chen, H. Matano and L. Véron: Anisotropic singularities of solutions of nonlinear elliptic equations in R2, J. Differential Functional Analysis, 83 (1989), 50–97.

    Article  MATH  Google Scholar 

  10. S.-N. Chow, X.-B. Lin and K. Lu: Smooth foliations for flows in a Banach space, preprint.

    Google Scholar 

  11. S.-N. Chow and K. Lu: C k centre unstable manifold, Proc. Royal Soc. Edinburgh, 108A (1988), 303–320.

    Article  MathSciNet  Google Scholar 

  12. D. Gilbarg and N.S. Trüdinger: Elliptic partial differential equations of second order (2nd ed.), Springer-Verlag, Berlin/New York, 1977.

    Book  MATH  Google Scholar 

  13. J.K. Hale: Asymptotic behavior of dissipative systems, Math. Surveys and Monographs, 25, Amer. Math. Soc., Providence, R. I., 1988.

    MATH  Google Scholar 

  14. J.K. Hale, L.T. Magalhães and W.M. Oliva: An introduction to infinite dimensional dynamical systems — Geometric theory, Appl. Math. Sci., 47, Springer Verlag, New York, 1984.

    Book  MATH  Google Scholar 

  15. D. Henry: Geometric theory of semilinear parabolic equations, Lecture Notes in Math., 840, Springer Verlag, New York, 1981.

    MATH  Google Scholar 

  16. D. Henry: Some infinite-dimensiona Morse-Smale systems defined by parabolic differential equations, J. Differential Equations, 59 (1985), 165–205.

    Article  MathSciNet  MATH  Google Scholar 

  17. M.W. Hirsh, C.C. Pugh and M. Shub: Invariant manifolds, Lecture Notes in Math., 583, Springer Verlag, New York, 1977.

    Book  MATH  Google Scholar 

  18. K. Kirchgässner: Nonlinear wave motion and homoclinic bifurcation, Theoretical and Applied Mechanics (edits, F. Noirdson and N. Olhoff), Elsevier Science Publishers B. V. (North-Holland), 1985.

    Google Scholar 

  19. C. Loewner and L. Nirenberg: Partial differential equations invariant under conformal or projective transformations, Contributions to Analysis (edits, L.V. Ahlfors et al), Academic Press, Orlando, 1974, 245–272.

    Google Scholar 

  20. H. Matano: Existence of nontrivial unstable sets for equilibriums of strongly order-preserving systems, J. Fac. Sci. Univ. Tokyo, 30 (1983), 645–673.

    MathSciNet  MATH  Google Scholar 

  21. H. Matano: Correction to: Existence of nontrivial unstable sets for equilibriums of strongly order preserving systems, J. Fac. Sci. Univ. Tokyo, 34 (1987) 853–855.

    MathSciNet  MATH  Google Scholar 

  22. H. Matano: Nonlinear elliptic equations and infinite-dimensional dynamical systems, in preparation.

    Google Scholar 

  23. A. Mielke: A reduction principle for nonautonomous systems in infinite dimensional spaces, J. Differential Equations, 65 (1986), 68–88.

    Article  MathSciNet  MATH  Google Scholar 

  24. R. Osserman: On the inequality Δu ≥ f(u), Pacific J. Math., 7 (1957), 1641–1647.

    Article  MathSciNet  MATH  Google Scholar 

  25. J. Palis and W. de Melo: Geometric theory of dynamical systems, Springer Verlag, New York, 1980.

    MATH  Google Scholar 

  26. L. Simon: Asymptotics for a class of nonlinear evolution equations with applications to geometric problems, Ann. Math., 118 (1983), 525–571.

    Article  MathSciNet  MATH  Google Scholar 

  27. L. Véron: Singular solutions of some nonlinear elliptic equations, Nonlinear Anal, 5 (1981), 225–242.

    Article  MathSciNet  MATH  Google Scholar 

  28. L. Véron: Global behaviour and symmetry properties of singular solutions of nonlinear elliptic equations, Ann. Fac. Sci. Toulouse Math., 6 (1984), 1–31.

    Article  MathSciNet  MATH  Google Scholar 

  29. H.-O. Walther: Inclination lemmas with dominated convergence, Z. Angew. Math. Phys., 38 (1987), 327–337.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Tokyo, 113, Hongo, Tokyo, Japan

    Hiroshi Matano

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  1. Hiroshi Matano
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Hiroshi Fujita Teruo Ikebe Shige Toshi Kuroda

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© 1990 Springer-Verlag

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Matano, H. (1990). Singular solutions of a nonlinear elliptic equation and an infinite dimensional dynamical system. In: Fujita, H., Ikebe, T., Kuroda, S.T. (eds) Functional-Analytic Methods for Partial Differential Equations. Lecture Notes in Mathematics, vol 1450. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0084899

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  • DOI: https://doi.org/10.1007/BFb0084899

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  • Print ISBN: 978-3-540-53393-1

  • Online ISBN: 978-3-540-46818-9

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