Mathematical Sciences: Topics on Free Boundary Problems and Singular Parabolic Equations

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This project will focus on local and asymptotic behavior of nonlinear degenerate or singular evolution equations. Local regularity, appearance of singularities, and intrinsic local and global versions of Harnack type of estimates will be attempted. It is anticipated that questions concerning new type of weak solutions and intriguing aspects of parabolic equations with limiting Sobolev exponent will have to be analyzed. The equations to be studied arise as models of nonlinear phenomena such as fast diffusion in plasma, transition of phase, and flow of immiscible fluids in a porus medium. Problems concerning coating of metal surfaces with colloidal fluids and conduction-convection will be considered.