Existence of periodic solutions for semilinear parabolic equations

Norimichi Hirano; Noriko Mizoguchi

Banach Center Publications (1996)

  • Volume: 35, Issue: 1, page 39-49
  • ISSN: 0137-6934

Abstract

top
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 + × 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 → ±∞.

How to cite

top

Hirano, Norimichi, and Mizoguchi, Noriko. "Existence of periodic solutions for semilinear parabolic equations." Banach Center Publications 35.1 (1996): 39-49. <http://eudml.org/doc/251308>.

@article{Hirano1996,
abstract = {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 → ±∞.},
author = {Hirano, Norimichi, Mizoguchi, Noriko},
journal = {Banach Center Publications},
keywords = {multiplicity of T-periodic solutions},
language = {eng},
number = {1},
pages = {39-49},
title = {Existence of periodic solutions for semilinear parabolic equations},
url = {http://eudml.org/doc/251308},
volume = {35},
year = {1996},
}

TY - JOUR
AU - Hirano, Norimichi
AU - Mizoguchi, Noriko
TI - Existence of periodic solutions for semilinear parabolic equations
JO - Banach Center Publications
PY - 1996
VL - 35
IS - 1
SP - 39
EP - 49
AB - 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 → ±∞.
LA - eng
KW - multiplicity of T-periodic solutions
UR - http://eudml.org/doc/251308
ER -

References

top
  1. [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. Zbl0698.35075
  2. [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. [3] A. Beltramo and P. Hess, On the principal eigenvalue of a periodic-parabolic operator, Comm. Part. Diff. Eq. 9 (1984), 919-941. Zbl0563.35033
  4. [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. Zbl0426.35038
  5. [5] D. Henry, Geometric theory of semilinear parabolic equaitons, Lecture Notes in Math. 840, Springer-Verlag, New York, 1981. 
  6. [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. [7] N. Hirano, Existence of multiple periodic solutions for a semilinear evolution equations, Proc. Amer. Math. Soc. 106 (1989), 107-114. Zbl0729.35006
  8. [8] N. Hirano, Existence of nontrivial solutions of semilinear elliptic equaitons, Nonlinear Anal. 13 (1989), 695-705. Zbl0735.35055
  9. [9] N. Hirano, Existence of unstable periodic solutions for semilinear parabolic equations, to appear in Nonlinear Analysis. Zbl0814.35058
  10. [10] M. W. Hirsch, Differential equations and convergence almost everywhere in strongly monotone semiflows, Contemporary Math. 17 (1983), 267-285. Zbl0523.58034
  11. [11] J. Prüss, Periodic solutions of semilinear evolution equations, Nonlinear Anal. 3 (1979), 601-612. Zbl0419.34061
  12. [12] I. I. Vrabie, Periodic solutions for nonlinear evolution equations in a Banach space, Proc. Amer. Math. Soc. 109 (1990), 653-661. Zbl0701.34074

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.