一类非线性热方程的时间周期解

Time Periodic Solution to A Nonlinear Heat Equation

考虑一个边界上具有活塞项的非线性热方程的时间周期解问题,其中活塞项是一个时间周期函数.在过去的几十年,含有各种非线性源项的非线性扩散方程的齐次Dirichlet边值或Neumann边值问题的研究已经取得了丰富的成果,但对含有时间周期边界问题的研究很少.文中分别考虑了次线性、线性以及超线性情形下的周期解存在性,利用不动点方法和拓扑度方法,首先对次线性、线性情形,对任意的边值证明了大时间周期解的存在性;而对超线性情形,证明了当边值适当小时时间周期解的存在性.

A nonlinear heat equation with a piston on the boundary is discussed,where the piston term is a timeperiodic function. In the past several decades,the study of time periodic solutions for all kinds nonlinear diffusion equations with nonlinear sources have achieved fruitful results,but there are very few results for the study of the diffusion equations with time periodic boundary. The linear,sublinear and superlinear cases are studied respectively,and it is shown that for the sublinear case and linear case,there always exists time periodic solution for any piston;while for the superlinear case,time periodic solution exists when the boundary value is small.