On the Łojasiewicz-Simon gradient inequality

Journal of Functional Analysis

Volumn 201, Issue 2, 2003, Pages 572-601

  Chill, Ralph ^a  

^a UNIVERSITY OF ULM   (Germany)

Author keywords

Asymptotic behaviour; Cahn Hilliard equation; Convergence to equilibrium; Gradient like evolution equation; Kirchhoff Carrier equation; Quansilinear equation; Semilinear diffusion equation; Semilinear wave equation; ojasiewicz Simon inequality

Indexed keywords

EID: 0038116267     PISSN: 00221236     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0022-1236(02)00102-7     Document Type: Article

References (35)

1. B. Aulbach, Continuous and Discrete Dynamics near Manifolds of Equilibria, in: Lecture Notes in Mathematics, Vol. 1058, Springer, Berlin, New York, Heidelberg, 1984.
2. Y. Belaud, B. Helffer, L. Véron, Long-time vanishing properties of solutions of some semilinear parabolic equations, Ann. Inst. Henri Poincaré, Anal. Nonlinéaire 18 (2001) 43-68.
3. P. Brunovský, P. Polačik, On the local structure of ω-limit sets of maps, Z. Angew. Math. Phys. 48 (1997) 976-986.
4. R. Chill, M.A. Jendoubi, Convergence to steady states in asymptotically autonomous semilinear evolution equations, Nonlinear Anal. Ser. A: Theory Methods, 2003, to appear.
5. D. Daners, Robin boundary value problems on arbitrary domains, Trans. Amer. Math. Soc. 352 (2000) 4207-4236.
6. R. Dautray, J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques, Vol. VI, INSTN: Collection Enseignement, Masson, Paris, 1988.
7. E. Feireisl, F. Simondon, Convergence for semilinear degenerate parabolic equations in several space dimensions, J. Dynam. Differential Equations 12 (2000) 647-673.
8. E. Feireisl, P. Takač, Long-time stabilization of solutions to the Ginzburg-Landau equations of superconductivity, Monatsh. Math. 133 (2001) 197-221.
9. J.K. Hale, G. Raugel, Convergence in gradient-like systems with applications to PDE, Z. Angew. Math. Phys. 43 (1992) 63-124.
10. A. Haraux, M.A. Jendoubi, Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity, Calc. Var. Partial Differential Equations 9 (1999) 95-124.
11. A. Haraux, M.A. Jendoubi, Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity, Asymptotic Anal. 26 (2001) 21-36.
12. A. Haraux, M.A. Jendoubi, On the convergence of global and bounded solutions of some evolution equations, preprint, 2001.
13. A. Haraux, M.A. Jendoubi, O. Kavian, Rate of decay to equilibrium in some semilinear parabolic equations, J. Evolution Equations, 2003, to appear.
14. K.-H. Hoffmann, P. Rybka, Convergence of solutions to Cahn-Hillard equation, Commun. Partial Differential Equations 24 (1999) 1055-1077.
15. S.-Z. Huang, P. Takač, Convergence in gradient-like systems which are asymptotically autonomous and analytic, Nonlinear Anal. Ser. A: Theory Methods 46 (2001) 675 698.
16. M.A. Jendoubi, Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity, J. Differential Equations 144 (1998) 302-312.
17. M.A. Jendoubi, A simple unified approach to some convergence theorems of L. Simon, J. Funct. Anal. 153 (1998) 187-202.
18. M.A. Jendoubi, P. Polačik, Nonstabilizing solutions of semilinear hyperbolic and elliptic equations with damping, Prépublications de l'Université de Versailles-St. Quentin, 74, 2001.
19. f-stratification of subanalytic functions and the Łojasiewicz inequality, C. R. Acad. Sci. Paris, Sér. I 318 (1994) 129-133.
20. S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels, Colloques internationaux du C.N.R.S.: Les équations aux dérivées partielles, Paris (1962), Editions du C.N.R.S., Paris, 1963, pp. 87-89.
21. S. Łojasiewicz, Ensembles semi-analytiques, Preprint, I.H.E.S., Bures-sur-Yvette, 1965.
22. S. Łojasiewicz, Sur les trajectoires du gradient d'une fonction analytique, Seminari di Geometria, Bologna (1982/83), Universitá degli Studi di Bologna, Bologna, 1984, pp. 115-117.
23. S. Łojasiewicz, M.A. Zurro, On the gradient inequality, Bull. Polish Acad. Sci. Math. 47 (1999) 143-145.
24. H. Matano, Convergence of solutions of one-dimensional semilinear heat equations, J. Math. Kyoto Univ. 18 (1978) 221-227.
25. P. Mironescu, V. Radulescu, Nonlinear Sturm-Liouville type problems with finite number of solutions, preprint, 1999.
26. J. Palis, W. de Melo, Geometric Theory of Dynamical Systems. An Introduction, Springer, New York, Heidelberg, Berlin, 1982.
27. P. Polačik, K. Rybakowski, Nonconvergent bounded trajectories in semilinear heat equations, J. Differential Equations 124 (1996) 472-494.
28. P. Polačik, F. Simondon, Nonconvergent bounded solutions of semilinear heat equations on arbitrary domains, preprint, 2001.
29. M. Reed, B. Simon, Methods of Modern Mathematical Physics, I: Functional Analysis, 2nd edition, Academic Press, New York, 1980.
30. L. Simon, Asymptotics for a class of non-linear evolution equations, with applications to geometric problems, Ann. Math. 118 (1983) 525-571.
31. L. Simon, Theorems on Regularity and Singularity of Energy Minimizing Maps, Lecture Notes in Mathematics ETH Zürich, Birkhäuser, Basel, 1996.
32. K. Taira, A mathematical analysis of thermal explosions, Int. J. Math. Mech. Sci. 28 (2001) 581-607.
33. K. Taira, K. Umezu, Stability in chemical reactor theory, in: G. Lumer, L. Weis (Eds.), Sixth Conference on Evolution Equations and Their Applications in Life Sciences, Bad Herrenalb (1998), Lecture Notes in Pure and Applied Mathematics, Vol. 215, Marcel Dekker, New York, 2000, pp. 421-433.
34. E. Zeidler, Nonlinear Functional Analysis and Its Applications. I, Springer, New York, Berlin, Heidelberg, 1990.
35. T.I. Zelenyak, Stabilization of solutions of boundary value problems for a second-order parabolic equation with one space variable, Differential Equations 4 (1968) 17-22 (Transl. from Differ. Uravn. 4 (1968) 34-45).