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

中文翻译：

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Chafee-Infante 方程解的渐近行为

在更高维度上，非自治 Chafee-Infante 方程的动力学有许多有趣且具有挑战性的问题。本文关注的是非自治 Chafee-Infante 方程 $$\frac{\partial u}{\partial t}-\Delta u = \lambda(t)(uu^3)$$ 的解的渐近行为∂ u ∂ t − Δ u = λ ( t ) ( u − u 3 ) 在更高维度，其中 λ ( t ) ∈ C 1 [0, T ] 和 λ ( t ) 是一个正的周期函数。我们将 λ 1 表示为 − △ ϕ = λ ϕ, x ∈ Ω 的第一个特征值；ϕ = 0, x ∈ ∂ Ω。对于任何空间维度 N ≥ 1，我们证明如果 λ ( t ) ≤ λ 1 ，那么非平凡解收敛到零，即 $$\lim_{t \rightarrow +\infty}$$ lim t → + ∞ u ( x , t ) = 0, x ∈ Ω; 如果 λ ( t ) > λ 1 as t → +∞，则正解被正周期解“吸引”。特别，如果 λ ( t ) 与 t 无关，则正解收敛于 −△ U = λ ( U − U 3 ) 的正解。此外，还提供了数值模拟来验证我们的结果。