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某些爆破性质，这给整体解、爆破解和数值解带来了许多具有挑战性的问题。本研究的目的是使用一种新的泛函方法和移动网格方法从理论和数值两方面研究整体解和爆破解的性质，包括与抛物系统定态有关的椭圆组。预期结果将包括： 1.引入了一种新的泛函方法来讨论物理中的多孔介质模型、化学中的趋化性模型和生态学中的互惠模型的整体解和爆破解。研究了系统正平衡态的存在性、唯一性和稳定性。 2.研究了一类椭圆方程组和分支曲线的多个正解的存在性。 3.对已有的移动网格法和其他自适应网格法进行了改进，使之能够数值求解一些复杂的方程或系统，如解在空间无穷远处爆破的方程。给出了具有守恒质量和能量的薛定谔方程以及无界域的其他方程的移动网格格式。 4.研究了一类更一般的拟线性抛物型方程组，得到了初值条件下整体解存在的充分条件，理论上和数值上的爆破性质，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.