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## Banach Center Publications

1996 | 35 | 1 | 39-49
Tytuł artykułu

### Existence of periodic solutions for semilinear parabolic equations

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In this paper, we are concerned with the semilinear parabolic equation ∂u/∂t - Δu = g(t,x,u) if $(t,x) ∈ R_{+} × Ω$ u = 0 if $(t,x) ∈ R_{+} × ∂Ω$, where $Ω ⊂ R^{N}$ is a bounded domain with smooth boundary ∂Ω and $g : R _{+} × \bar{Ω} × R → R$ is T-periodic with respect to the first variable. The existence and the multiplicity of T-periodic solutions for this problem are shown when g(t,x,ξ)/ξ lies between two higher eigenvalues of - Δ in Ω with the Dirichlet boundary condition as ξ → ±∞.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
39-49
Opis fizyczny
Daty
wydano
1996
Twórcy
autor
• Department of Mathematics, Faculty of Engineering, Yokohama National University, Tokiwadai, Hodogaya-ku, Yokohama 156, Japan
autor
• Department of Information Science, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152, Japan
Bibliografia
• [1] N. D. Alikakos, P. Hess and H. Matano, Discrete order preserving semigroups and stability for periodic parabolic differential equaitons, J, Diff. Eq. 82 (1989), 322-341.
• [2] H. Amann, Periodic solutions for semi-linear parabolic equations, in 'Nonlinear Analysis: A Collection of Papers in Honor of Erich Rothe', Academic Press, New York, 1978, 1-29.
• [3] A. Beltramo and P. Hess, On the principal eigenvalue of a periodic-parabolic operator, Comm. Part. Diff. Eq. 9 (1984), 919-941.
• [4] A. Castro and A. Lazer, Critical point theory and the number of solutions of a Dirichlet problem, Ann. Math. Pure Appl. 70 (1979), 113-137.
• [5] D. Henry, Geometric theory of semilinear parabolic equaitons, Lecture Notes in Math. 840, Springer-Verlag, New York, 1981.
• [6] P. Hess, On positive solutions of semilinear periodic-parabolic problems in infinite-dimensional systems, ed. Kappel-Schappacher, Lecture Notes in Math. 1076 (1984), 101-114.
• [7] N. Hirano, Existence of multiple periodic solutions for a semilinear evolution equations, Proc. Amer. Math. Soc. 106 (1989), 107-114.
• [8] N. Hirano, Existence of nontrivial solutions of semilinear elliptic equaitons, Nonlinear Anal. 13 (1989), 695-705.
• [9] N. Hirano, Existence of unstable periodic solutions for semilinear parabolic equations, to appear in Nonlinear Analysis.
• [10] M. W. Hirsch, Differential equations and convergence almost everywhere in strongly monotone semiflows, Contemporary Math. 17 (1983), 267-285.
• [11] J. Prüss, Periodic solutions of semilinear evolution equations, Nonlinear Anal. 3 (1979), 601-612.
• [12] I. I. Vrabie, Periodic solutions for nonlinear evolution equations in a Banach space, Proc. Amer. Math. Soc. 109 (1990), 653-661.
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Bibliografia
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