Grant: Discovery Grants Program - Individual

Quasilinear degenerate parabolic equations have many applications in Physics, Chemistry, Biology and image processing. These equations involve a degenerate or singular diffusion term and some kinds of blowup properties which create many challenging problems in global, blowup and numerical solutions. The objectives of this research program involve using a new functional method and moving mesh methods to investigate the properties of global and blowup solutions both theoretically and numerically. The expected results will include the following: 1. Introduce a new functional method to discuss the global and blowup solutions for a system of a porous medium type with lower order terms and the strongly coupled systems in which the diffusion terms involve degenerate and derivative terms. 2. Investigate the properties of the parabolic equations related to image processing which includes noise reduction, anisotropic filtering, image segmentation, registration and deformation, color image in-painting, etc. 3. Modify the existing algorithms of moving mesh methods and other adaptive grid methods to apply to numerical computations of some complicated equations, such as the models of image processing. 4. Deal with a class of more general quasi-linear parabolic equations including chemotaxis model to find sufficient conditions on the initial conditions for the global existence and blowup properties both theoretically and numerically.