Existence of periodic solutions for semilinear parabolic equations

Volume 35 / 1996

Norimichi Hirano, Noriko Mizoguchi Banach Center Publications 35 (1996), 39-49 DOI: 10.4064/-35-1-39-49


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 → ±∞.


  • Norimichi Hirano
  • Noriko Mizoguchi

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