非局部时滞扩散系统的行波解和整体解

项目名称： 非局部时滞扩散系统的行波解和整体解

项目编号： No.10871085

项目类型： 面上项目

立项/批准年度： 2009

项目学科： 金属学与金属工艺

项目作者： 李万同

作者单位： 兰州大学

项目金额： 29万元

中文摘要： 本项目利用单调动力系统原理、偏微分方程理论和非线性泛函分析技巧,研究了空间扩散模型的空间传播机制,利用行波解、整体解、渐近传播以及Turing模式等指标体现了空间非局部性、时滞以及交错扩散的非平凡影响。在行波解存在性与定性理论研究中，我们得到了行波解存在性、稳定性和唯一性等结果，这些结论体现出了空间非均匀性和非局部性、高维空间等因素对行波传播方向的本质影响；在整体解研究中，构造出了新型整体解并研究了整体解对于传播方向和传播速度的依赖性等问题，这些整体解从理论上对于理解全局吸引子结构和系统瞬间动力学非常有用，在应用中可以很好的描述不能用行波解描述的传播现象；在渐近传播理论研究中，得到了竞争系统传播速度以及传播效果的一些估计，体现出了描述耦合非线性在渐近传播中的效果，从理论上证明了种间竞争在空间传播中的负面作用；利用单调动力系统以及分支理论，研究了一些经典模型的长时间行为以及稳态解性质，得到了Turing模式以及一些新型稳态解存在性。这些结果可以描述已有结论不能描述的一些自然现象。

中文关键词： 整体解；行波解；渐近传播；Turing模式与分支；单调动力系统

英文摘要： This project is concerned with the propagation patterns of diffusion models by the principle of monotone dynamical systems and theory of partial differential equations and methods of nonlinear functional analysis, during which the nontrivial role of spatial nonlocality, time delay and cross diffusion is formulated by traveling wave solutions, entire solutions, asymptotic spreading and Turing patterns. In the study of traveling wave solutions as well as their qualitative analysis, we proved the existence, asymptotic stability and uniqueness of traveling wave solutions, our results reflected the essential effect of spatial inhomogeneity, spatial nonlocality and higher dimension by the direction of propagation. We also constructed new type entire solutions of evolutionary systems and discussed the parameters dependence of propagation directions and wave speeds, which was very useful in the understanding of structure of global attractors and instantaneous behavior of dynamical systems. In particular, these entire solutions formulated many phenomena that could not be reflected by well studied traveling wave solutions. For the asymptotic spreading of a competitive system, we obtained some estimates of two competitive invaders by asymptotic speed of spreading as well as propagation consequence, which implied the negative effect of interspecific competition from the viewpoint of spatial propagation. Using the theory of monotone dynamical systems and bifurcation method, the long time behavior and steady states of some classical models were also investigated, which indicated the Turing patterns and some new type steady states. Our conclusions could illustrate many important nature phenomena which could not be formulated by the known results.

英文关键词： Entire solutions；traveling wave solutions；asymptotic spreading；Turing patterns and bifurcation；monotone dynamical systems