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 ${\displaystyle (t,x)\in R_{+}\times \Omega }$ u = 0 if ${\displaystyle (t,x)\in R_{+}\times \partial \Omega }$, where ${\displaystyle \Omega \subset R^N}$ is a bounded domain with smooth boundary ∂Ω and ${\displaystyle g:R_{+}\times \overline{\Omega }\times R\to R}$ 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 → ±∞.