ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR THE CHAFEE-INFANTE EQUATION

In higher dimension,there are many interesting and challenging problems about the dynamics of non-autonomous Chafee-Infante equation.This article is concerned with the asymptotic behavior of solutions for the non-autonomous Chafee-Infante equation∂u∂t−Δu=λ(t)(u−u3)in higher dimension,whereλ(t)∈C1[0,T]andλ(t)is a positive,periodic function.We denoteλ1 as the first eigenvalue of−Δφ=λφ,x∈Ω;φ=0,x∈∂Ω.For any spatial dimension N≥1,we prove that ifλ(t)≤λ1,then the nontrivial solutions converge to zero,namely,limt→+∞u(x,t)=0,x∈Ω;ifλ(t)>λ1 as t→+∞,then the positive solutions are"attracted"by positive periodic solutions.Specially,ifλ(t)is independent of t,then the positive solutions converge to positive solutions of−ΔU=λ(U−U^3).Furthermore,numerical simulations are presented to verify our results.