Existence of periodic solutions for semilinear parabolic equations

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 → ±∞.