具有时空周期性和奇异性反应扩散方程若干问题的研究

项目名称： 具有时空周期性和奇异性反应扩散方程若干问题的研究

项目编号： No.11271167

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 彭锐

作者单位： 江苏师范大学

项目金额： 60万元

中文摘要： 本项目拟研究生态学和传染病学中有一些具有实际应用的反应扩散方程（组）。 这些方程具有时空周期性和奇异性参数。模型更加接近实际环境，在以往的研究工作中很少涉及。我们拟研究问题解的各种性态，包括正周期解的存在性、不存在性、唯一性、多解性、稳定性、关于参数的渐近行为以及初边值问题的动力学结构等。 主要研究对象是单个Logistic方程、生态学中的竞争和捕食模型，以及典型的传染病模型等。通过发展新的数学思想和技巧，我们试图揭示时空周期性、奇异性和退化性环境、扩散方式对于生态物种和传染病的时空分布、灭绝或者共存等性态的本质影响。深入探讨这些模型理论定性性质，一方面在此过程中寻求解决问题的数学理论和技巧，可以有力地推动偏微分方程研究理论自身发展。另一方面, 通过深入理解物种、疾病的时空发展过程, 揭示、预测其变化趋势, 从而寻求预防和控制、保护等的最优策略，具有潜在的应用价值。

中文关键词： 反应扩散方程；时空奇异性；周期性；退化性；解

英文摘要： In this project, we plan to study some nonlinear reaction-diffusion equations mainly arising from biology and epidemiology. These equations， which were studied previously either in the autonomous case or in the nondegenerate case, incorporate spatiotemporal periodicity, heterogeneity and/or degeneracy so that they are more realistic in describing the evolution of species and infectious diseases. By developing new mathematical ideas and techniques, we will study various properties of positive periodic solutions to these PDE problems such as the existence, non-existence, uniqueness, multiplicity, stability and asymptotic behavior of positive solutions with respect to certain parameters and their dynamics with respect to the associated initial-boundary value problems, and explore essential impacts of the environmental factors on their dynamical behaviors such as persistence and extinction. The problems to be studied in the project include the scalar logistic equation, the competition and predator-prey models from ecology as well as important equations from epidemiology. Our investigation will induce new mathematical ideas and techniques which thereby will considerably contribute to the development of PDE theory; on the other hand, our theoretical and numerical results will help to enhance our deeper understanging o

英文关键词： reaction-diffusion equation；spatiotemporal heterogeneity；periodicity；degeneracy；solution