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Existence of periodic solutions for semilinear parabolic equations

Norimichi Hirano; Noriko Mizoguchi — 1996

Banach Center Publications

In this paper, we are concerned with the semilinear parabolic equation ∂u/∂t - Δu = g(t,x,u) if $(t, x) \in R_{+} \times Ω$ u = 0 if $(t, x) \in R_{+} \times \partial Ω$ , where $Ω \subset R^{N}$ is a bounded domain with smooth boundary ∂Ω and $g : R_{+} \times \bar{Ω} \times R \to R$ 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 → ±∞.

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