The Infinite Dimensional Case

  • Alain HarauxEmail author
  • Mohamed Ali Jendoubi
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)


This second core chapter concerns the convergence problem in infinite dimensions. In this framework, analyticity of the potential no longer implies the Łojasiewicz gradient inequality. We devise an essentially optimal condition for this inequality to hold. Applications to semi linear PDE of parabolic or hyperbolic type are given as an application of general theorems.


  1. 1.
    L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems. Ann. Math. (2) 118(3), 525–571 (1983)Google Scholar
  2. 2.
    T.I. Zelenyak, Stabilization of solutions of boundary value problems for a second-order parabolic equation with one space variable. (Russian) Differencialnye Uravnenija 4, 34–45 (1968)Google Scholar
  3. 3.
    H. Matano, Convergence of solutions of one-dimensional semilinear parabolic equations. J. Math. Kyoto Univ. 18, 221–227 (1978)MathSciNetzbMATHGoogle Scholar
  4. 4.
    A. Haraux, M.A. Jendoubi, The Łojasiewicz gradient inequality in the infinite-dimensional Hilbert space framework. J. Funct. Anal. 260, 2826–2842 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    A. Haraux, M.A. Jendoubi, Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity. Calc. Var. Partial Diff. Equat. 9(2), 95–124 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    O. Kavian, Introduction la théorie des points critiques et applications aux problèmes elliptiques. Mathématiques & applications (Berlin), vol. 13. (Springer, Paris, 1993)Google Scholar
  7. 7.
    S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Comm. Pure Appl. Math. 12, 623–727 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    A. Haraux, M. Kirane, Estimations \(C^1\) pour des problèmes paraboliques semi-linéaires. Ann. Fac. Sci. Toulouse Math. 5, 265–280 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    A. Haraux, M.A. Jendoubi, On the convergence of global and bounded solutions of some evolution equations. J. Evol. Equat. 7(3), 449–470 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    J.K. Hale, G. Raugel, Convergence in gradient-like systems with applications to PDE. Z. Angew. Math. Phys. 43, 63–124 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    A. Haraux, P. Poláčik, Convergence to a positive equilibrium for some nonlinear evolution equations in a ball. Acta Math. Univ. Comenian. (N.S.) 61, 129–141 (1992)Google Scholar
  12. 12.
    P. Brunovský, P. Poláčik, On the local structure of -limit sets of maps. Z. Angew. Math. Phys. 48, 976–986 (1997)MathSciNetzbMATHGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.Laboratoire Jacques-Louis LionsSorbonne Universités, UPMC Univ Paris 06, CNRS, UMR 7598ParisFrance
  2. 2.Institut Préparatoire aux Etudes Scientifiques et TechniquesUniversité de CarthageLa MarsaTunisia
  3. 3.Faculté des sciences de Tunis, Laboratoire EDP-LR03ES04Université de Tunis El ManarTunisTunisia

Personalised recommendations