Grant: Discovery Grants Program - Individual

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.