In higher dimension,there are many interesting and challenging problems about the dynamics of non-autonomous Chafee-Infante equation. This article is conced with the asymptotic behavior of solutions for the non–autonomous Chafee-Infante equation(?u/?t)??u=λ(t)(u?u3) in higher dimension,whereλ(t)∈C1[0,T ] andλ(t) is a positive,periodic function. We denote λ1 as the first eigenvalue of ???=λ?,x∈?; ?=0,x∈??. For any spatial dimension N ≥1,we prove that if λ(t)≤λ1,then the nontrivial solutions converge to zero,namely,(lim t→+∞u)(x,t) = 0,x ∈ ?; if λ(t)>λ1 as t → +∞,then the positive solutions areattractedby positive periodic solutions. Specially,if λ(t) is independent of t,then the positive solutions converge to positive solutions of ??U = λ(U ?U3). Furthermore,numerical simulations are presented to verify our results.