Existence and blowup of positive solutions for nonlinear elliptic and parabolic systems and their numerical computations

非线性椭圆抛物线系统正解的存在性、爆炸性及其数值计算

• 批准号: RGPIN-2014-03857
• 负责人: Chen, Shaohua
• 金额: $ 0.8万
• 依托单位: Cape Breton University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=647381
• 关键词: Existence; blowup; positive; solutions; nonlinear

The quasi-linear parabolic systems have many applications in Physics, Chemistry, Biology and image processing. Those systems involve degenerate or singular diffusion terms and some kinds of blowup properties which cause many challenging problems for global, blowup and numerical solutions. The objectives of this research program are using a new functional method and moving mesh methods to investigate properties of global and blowup solutions both theoretically and numerically, including elliptic systems related to steady states of parabolic systems. The expected results will include:**1. Introduce a new functional method to discuss global and blowup solutions for systems of porous medium model in physics, Chemotaxis model in chemistry and mutualistic model in ecology. Also study the existence, uniqueness and stability of positive steady states to the systems.**2. Investigate the existence of multi positive solutions of some elliptic systems and bifurcation curves. **3. Modify the existing algorithms of moving mesh methods and other adaptive grid methods to numerically solve some complicated equations or systems, such as equations whose solutions blow up at space infinity. Also develop a moving mesh scheme for Schrodinger equations with conservative mass and energy and other equations whose domains are unbounded. **4. Deal with a class of more general quasi-linear parabolic systems to find sufficient conditions on the initial conditions for the global existence and blowup properties both theoretically and numerically, as well as convergence to steady states.

拟线性抛物线系统在物理、化学、生物学和图像处理方面有许多应用。这些系统涉及简并或奇异扩散项以及某些类型的爆炸特性，这给全局、爆炸和数值解决方案带来了许多具有挑战性的问题。该研究项目的目标是使用新的泛函方法和移动网格方法从理论上和数值上研究全局解和爆炸解的性质，包括与抛物线系统稳态相关的椭圆系统。预期结果将包括：**1。引入一种新的函数方法来讨论物理中多孔介质模型、化学中趋化模型和生态学中互惠模型系统的全局解和爆炸解。并研究系统正稳态的存在性、唯一性和稳定性。**2．研究一些椭圆系统和分岔曲线的多个正解的存在性。 **3.修改现有的移动网格方法和其他自适应网格方法的算法，以数值求解一些复杂的方程或系统，例如解在无限空间爆炸的方程。还为具有保守质量和能量的薛定谔方程以及域无界的其他方程开发移动网格方案。 **4.处理一类更一般的拟线性抛物线系统，以在理论和数值上找到全局存在和爆炸特性的初始条件的充分条件，以及收敛到稳态。