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

拟线性抛物系统在物理、化学、生物和图像处理等领域有着广泛的应用。这些系统涉及退化或奇异扩散项和一些爆破性质，这给全局解、爆破解和数值解带来了许多具有挑战性的问题。本研究计划的目标是使用新的泛函方法和移动网格方法，从理论和数值上研究全局解和爆破解的性质，包括与抛物系统稳态相关的椭圆系统。预期结果将包括：**1。介绍了一种新的泛函方法来讨论物理上的多孔介质模型、化学上的趋化性模型和生态学上的共生模型系统的全局解和爆破解。研究了系统正稳态的存在性、唯一性和稳定性。研究了一类椭圆系统和分岔曲线的多正解的存在性。* * 3。改进现有的移动网格法和其他自适应网格法的算法，对一些复杂的方程或系统进行数值求解，例如解在空间无穷远处爆炸的方程。同时，对具有保守质量和能量的薛定谔方程和其他无界方程也提出了一种运动网格格式。* * 4。处理一类更一般的拟线性抛物型系统，从理论上和数值上寻找其整体存在的初始条件、爆破性质以及收敛到稳态的充分条件。

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.