Asymptotic Behavior of Solutions for the Chafee-Infante Equation

Abstract

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 $$\frac{\partial u}{\partial t}-\Delta u = \lambda(t)(u-u^3)$$ in higher dimension, where λ(t) ∈ C1 [0, T] and λ(t) is a positive, periodic function. We denote λ1 as the first eigenvalue of −△ϕ = λϕ, x ∈ Ω; ϕ = 0, x ∈ ∂Ω. For any spatial dimension N ≥ 1, we prove that if λ(t) ≤ λ1, then the nontrivial solutions converge to zero, namely, $$\lim_{t \rightarrow +\infty}$$u(x, t) = 0, x ∈ Ω; if λ(t) > λ1 as t → +∞, then the positive solutions are “attracted” by 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.