数学与统计学院知识库

对于非线性项为点火型、单稳型和双稳型的 Heaviside 型初值问题, 方程的解都收敛到单个波前解. 然而对于混合型非线性项, 单个波前解已经不能更好地刻画方程解的渐近行为, 因此需要引入传播台阶来研究多稳型反应扩散方程. 目前, 对空间均匀和非均匀环境下多稳型方程波前型初值问题的长时间行为已有了重要研究进展. 而关于时间非自治反应扩散方程的研究结果却很少. 本文将考虑带 Heaviside 型初值的时间周期多稳型反应扩散方程的长时间行为, 建立传播台阶的存在性和收敛性. 本文的结构如下:

第一章主要介绍本文的研究背景和主要结果. 第二章将得到时间周期反应扩散方程脉动型行波解的存在性和收敛性. 为此首先给出半线性抛物方程的零数理论, 然后给出其 Omega - 极限集的定义, 并建立方程关于 Omega - 极限集的重要引理. 在此基础上, 得到方程脉动型行波解的存在性和收敛性. 第三章主要研究传播台阶的存在性和收敛性. 假设第二章得到的脉动型行波解为传播台阶的第一阶, 然后通过脉动型行波解间的迭代作用, 得到极小传播台阶的存在唯一性和收敛性.

In some problems with Heaviside type initial value such as combustion, mono-stable and bistable nonlinearities, the solutions of equation convergence to a single front. However, for some heterogeneous nonlinearity, the single traveling wave front cannot characterized the asymptotic behavior of the solutions. Therefore, it is necessary to introduce a propagating terrace to reveal the multi-stable reaction-diffusion equations. At present, Important research has been made on the long time asymptotic behavior of the multi-stable equations with front-like initial value in spatially homogeneous and inhomogeneous environment. However, there are few results on non-autonomous reaction-diffusion equations. In this paper, we consider a class of time-periodic multi-stable types equations with Heaviside initial value, and mainly obtain the existence and convergence to a propagating terrace. This paper is organized as follows.

We mainly state the research background and the main results of this paper in chapter one. In chapter two, we first exhibit the zero-number theory of semi-linear parabolic equations. Then we shall give the definition of  Omega - limit sets and it's important lemma.  On that basis, the existence and convergence to pulsating traveling waves will be following. We first assume that the first step is the pulsating traveling wave in chapter three. Then we shall use the iteration of pulsating traveling waves to prove the existence and convergence to a minimal propagating terrace.