Convergence to steady states in asymptotically autonomous semilinear evolution equations

Nonlinear Analysis, Theory, Methods and Applications

Volumn 53, Issue 7-8, 2003, Pages 1017-1039

  Chill, R ^a   Jendoubi, M A ^b  

^a UNIVERSITY OF ULM   (Germany)

^b UNIVERSITÉ DE VERSAILLES SAINT QUENTIN EN YVELINES   (France)

Author keywords

Analytic nonlinearity; Convergence; Evolution equation; Nonautonomous

Indexed keywords

NONLINEAR EQUATIONS; PROBLEM SOLVING;

STEADY STATES;

CONVERGENCE OF NUMERICAL METHODS;

EID: 0037410017     PISSN: 0362546X     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0362-546X(03)00037-3     Document Type: Article

References (29)

1. Acquistapace P. Evolution operators and strong solutions of abstract linear parabolic equations. Differential Integral Equations. 1:1988;433-457.
2. Acquistapace P., Terreni B. A unified approach to abstract linear non-autonomous parabolic equations. Rend. Sem. Mat. Univ. Padova. 78:1987;47-107.
3. Batty C.J.K., Chill R. Approximation and asymptotic behaviour of evolution families. Differential Integral Equations. 15:2002;477-512.
4. Brunovský P., Polačik P. On the local structure of. ω -limit sets of maps Z. Angew. Math. Phys. 48:1997;976-986.
5. Cazenave Th., Haraux A. An Introduction to Semilinear Evolution Equations. Oxford Lecture Series in Mathematics and its Applications. Vol. 13:1998;Oxford University Press, New York.
6. Hale J.K., Raugel G. Convergence in gradient-like systems with applications to PDE. Z. Angew. Math. Phys. 43:1992;63-124.
7. Haraux A. Systèmes Dynamiques Dissipatifs et Applications. 1990;Masson, Paris.
8. Haraux A., Jendoubi M.A. Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity. Calc. Var. Partial Differential Equations. 9:1999;95-124.
9. A. Haraux, M.A. Jendoubi, On the convergence of global and bounded solutions of some evolution equations, Prépublication du Laboratoire Jacques-Louis Lions R02-003, 2002, submitted for publication.
10. Haraux A., Polačik P. Convergence to a positive equilibrium for some nonlinear evolution equations. Acta Math. Univ. Comen. 61:1992;129-141.
11. Hoffmann K.-H., Rybka P. Convergence of solutions to Cahn-Hillard equation. Commun. Partial Differential Equations. 24:1999;1055-1077.
12. Huang S.-Z., Takač P. Convergence in gradient-like systems which are asymptotically autonomous and analytic. Nonlinear Anal., Ser. A, Theory Methods. 46:2001;675-698.
13. Jendoubi M.A. Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity. J. Differential Equations. 144:1998;302-312.
14. Jendoubi M.A. A simple unified approach to some convergence theorems of L. Simon. J. Funct. Anal. 153:1998;187-202.
15. M.A. Jendoubi, P. Polačik, Nonstabilizing solutions of semilinear hyperbolic and elliptic equations with damping, Proc. Royal Soc. Edinburgh (2003) in press.
16. Lions P.-L. Structure of the set of steady-state solutions and asymptotic behaviour of semilinear heat equations. J. Differential Equations. 53:1984;362-386.
17. S. Lojasiewicz, Ensembles semi-analytiques, Preprint, I.H.E.S. Bures-sur-Yvette, 1965.
18. Lunardi A. Analytic Semigroups and Optimal Regularity in Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications. Vol. 16:1995;Birkhäuser, Basel.
19. Matano H. Convergence of solutions of one-dimensional semilinear heat equations. J. Math. Kyoto Univ. 18:1978;221-227.
20. Ono K. On global existence and asymptotic stability of solutions of mildly degenerate dissipative nonlinear equations of Kirchhoff type. Asymptot. Anal. 16:1998;299-314.
21. Park J.Y., Bae J.J., Jung I.H. On existence of global solutions for the Carrier model with nonlinear damping and source term. Appl. Anal. 77:2001;305-318.
22. Polačik P., Rybakowski K. Nonconvergent bounded trajectories in semilinear heat equations. J. Differential Equations. 124:1996;472-494.
23. P. Polačik, F. Simondon, Nonconvergent bounded solutions of semilinear heat equations on arbitrary domains, preprint, 2001.
24. Simon L. Asymptotics for a class of non-linear evolution equations, with applications to geometric problems. Ann. Math. 118:1983;525-571.
25. L. Simon, Theorems on Regularity and Singularity of Energy Minimizing Maps, Lecture Notes in Mathematics ETH Zürich, Birkhäuser, Basel, 1996.
26. Tanabe H. Functional Analytic Methods for Partial Differential Equations. Monographs in Pure and Applied Mathematics. Vol. 204:1997;Marcel Dekker, New York, Basel, Hong Kong.
27. Webb G.F. Compactness of bounded trajectories of dynamical systems in infinite dimensional spaces. Proc. Roy. Soc. Edinburgh A. 84:1979;19-34.
28. Yagi A. Parabolic equations in which the coefficients are generators of infinitely differentiable semigroups II. Funkcial. Ekvac. 33:1990;139-150.
29. Zelenyak T.I. 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).