非局部扩散系统的行波解和整体解

项目名称： 非局部扩散系统的行波解和整体解

项目编号： No.11201359

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 孙玉娟

作者单位： 西安电子科技大学

项目金额： 22万元

中文摘要： 近年来非局部扩散方程开始广泛地应用于生态学、流行病学等学科的研究领域中。空间的非局部性不但导致了数学理论研究上的困难而且引起了动力学行为上的本质改变。因此，建立其系统理论是非常重要而有实际意义的课题。本课题计划研究非局部扩散方程组的整体解及其性质和高维空间中非局部扩散方程的行波解及相关性质。本项目将通过构造合适的上下解，运用比较原理建立方程组的整体解，并结合傅里叶变换等方法研究整体解关于参数的连续依赖性、稳定性等性质。拟利用最大值原理、平面滑动技术或比较原理证明方程行波解的存在性并获得相关性质。前期我们已经研究了一维空间中非局部扩散方程的行波解和整体解。本课题将完善和深化对非局部扩散系统动力学行为的研究，希望通过本课题的工作，为进一步理解高维空间中该系统的动力学行为提供理论依据。

中文关键词： 非局部扩散方程；整体解；非线性度；；

英文摘要： Recently, nonlocal dispersal equations have widely applied to ecosystem, epidemiology, materials sciences and so on. However, the effect of nonlocal dispersal not only leads to mathematical difficulties, but also essential changes of dynamics.Therefore, it is more meaningful and valuable in theory and practice to study such equations. This program will study entire solutions of nonlocal dispersal equation sets and taveling wave solutions of nonlocal dispersal equations in high dimensional space. The entire solutions of nonlocal dispersal equation sets will be established by constructing appropriate subsolutions and supersolutions and using comparison principle. In addition, the continuity, stability and so on of entire solutions will be obtained by Fourier transformation. The existence and some properties of traveling wave solutions will be obtained by maximal principle and spliding techniches or using comparison principle. Before this, we have studied traveling wave solutions and entire solutions of nonlocal dispersal equations in one-dimensional space. In this project, we will perfect and deepen the study of the dynamics of nonlocal dispesal systems, and hope to provide theoretical basis for deeply understanding the dynamics of this system in high dimensional space.

英文关键词： nonlocal dispersal equations；entire solutions；nonlinearity；；