关键词: Chafee-Infante equation, asymptotic behavior, periodic solutions

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 $\lambda(t)\in C^{1}[0,T]$ and $\lambda(t)$ is a positive, periodic function. We denote $\lambda_{1}$ as the first eigenvalue of $-\Delta \varphi = \lambda \varphi, \; x \in \Omega; \;\; \varphi=0, \; x \in \partial \Omega.$ For any spatial dimension $N\geq1$, we prove that if $\lambda(t)\leq\lambda_{1}$, then the nontrivial solutions converge to zero, namely, $\underset{t\rightarrow+\infty }{\lim} u(x,t) =0, \; x\in\Omega$; if $\lambda(t)>\lambda_{1}$ as $t\rightarrow +\infty$, then the positive solutions are ``attracted'' by positive periodic solutions. Specially, if $\lambda(t)$ is independent of $t$, then the positive solutions converge to positive solutions of $- \Delta U =\lambda(U -U^{3})$. Furthermore, numerical simulations are presented to verify our results.

Key words: Chafee-Infante equation, asymptotic behavior, periodic solutions