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

拟线性抛物型方程组在物理、化学、生物和图像处理等领域有着广泛的应用。这类方程组包含退化或奇异扩散项以及某些爆破性质，给整体解、爆破解和数值解带来了许多挑战性的问题。本研究计划的目标是使用一种新的功能方法和移动网格方法，从理论和数值上研究整体和爆破解决方案的性质，包括椭圆型系统相关的抛物型系统的稳定状态。预期的结果将包括：1。介绍了一种新的泛函方法来讨论物理学中的多孔介质模型、化学中的趋化模型和生态学中的互惠模型的整体解和爆破解。并研究了系统正平衡态的存在性、唯一性和稳定性.研究一类椭圆型方程组和分歧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，数值求解一些复杂的方程或系统，如方程的解在空间无穷远爆破。对于具有守恒质量和能量的薛定谔方程以及其他区域无界的方程，也发展了一种移动网格格式。4.研究一类更一般的拟线性抛物方程组，在初始条件下，从理论上和数值上得到方程组解的整体存在性、爆破性以及稳定性的充分条件.

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.