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.