格点动力系统收敛性和锐转变的研究

In this project we will consider lattice dynamical system, which is the spatially discrete counterpart of the classical reaction diffusion equation. We aim to study the convergence and sharp transitions for autonomous and time recurrent lattice systems by using a zero number argument, the reflection method, skew-product flow theory and the Floquet theory. Precisely, for autonomous lattice system, we will analyze phase portraits of the stationary equations, and prove that any bounded solution with compactly supported initial data converges to a stationary solution under the hypothesis that the nonlinearity is locally Lipschitz continuous. Then we will consider a monotone one-parameter family of initial data, and show that the transition between extinction and propagation is sharp for bistable nonlinearity. Also we will prove that for a class of KPP nonlinearity, there is exactly one value such that extinction occurs when the parameter does not exceed this value, otherwise propagation occurs. For time recurrent lattice system, by using an integer-valued Lyapunov function, we will establish the Floquet theory for linear system in the framework of skew-product semiflows, and prove the convergence and asymptotic symmetry properties for the bounded solution under the assumption that the nonlinearity is differentiable. Then for bistable nonlinearity, we will show some instability property of each threshold solution, leading eventually to the uniqueness of the threshold solution.

本项目考虑经典反应扩散方程对应的格点系统，旨在利用零点数论证、反射方法、斜积流理论、Floquet理论等动力系统的基本理论和最新成果，探讨自治和时间回复格点系统解的收敛性和锐转变。首先，对于自治格点系统，我们将分析其稳定态方程的相图，证明当非线性项满足局部Lipschitz连续性条件时，系统具有紧支初始结构有界解的收敛性。进而考虑单调依赖于某参数的一列初始值，对于双稳型非线性项，证明消失与传播之间的临界解对应唯一的参数；对于一类KPP型非线性项，证明当参数不超过某一特定值时，解的消失现象发生，否则解的传播现象发生。其次，对于时间回复格点系统，我们将在斜积半流的框架下结合整值Lyapunov函数的性质建立线性系统的Floquet理论，证明当非线性项满足可微性条件时，系统具有紧支初始结构有界解的收敛性和渐近对称性。进而针对双稳型非线性项，证明任一临界解的不稳定性，以得到临界解的唯一性。

本项目考虑经典反应扩散方程对应的格点系统，利用动力系统的基本理论和最新成果，探讨一般时间依赖（包括自治、时间回复）和随机格点方程解的渐近性态以及行波解的存在性、稳定性。首先，对于一般时间依赖和随机KPP型格点方程，我们通过建立比较原理和构造合适的上下解，研究了其渐近传播速度并证明了行波解的存在性、唯一性。同时，我们研究了系统解的长时间动力学行为，得到了行波解的渐近稳定性。其次，对于一般时间依赖和随机多种群竞争格点系统，我们建立了系统的比较原理，并通过构造合适的上下解，证明了其行波解的存在性和不存在性。我们指出，存在一个精确的阈值，行波解存在当且仅当其传播速度的下平均值大于此阈值。我们还根据国内外动力系统的发展前沿，在项目执行期间对海洋流体动力学中的稳态海洋水波的存在性及稳定性/不稳定性、浅水波Camassa-Holm方程的特征值及其最优估计等重要问题开展了探索性研究，并取得了一些有意义的成果。