Nonlinear Degenerate Parabolic Problems and Related Topics

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DMS-9801304 P. Daskalopoulos The abstact follows : The major objectives of this proposal are concerned with the study of certain quasilinear or fully-nonlinear degenerate parabolic equations in connection with more complexproblems of differential geometry, including the Ricci flow and the Gauss curvature flow. These problems find physical applications in such areas as population dynamics, diffusion in porous media, and thin liquid film dynamics. One specific area which will be investigated is the regularity question in free-boundary problems arising from the degeneracy of certain non-linear parabolic equations, including the porous medium equation, the evolution p-laplacian equation and Gauss Curvature flow with flat sides. Another specific area of the main objectives concerns the study of ultra-fast diffusion parabolic equations, which have received large attention because of their relationship to differential geometry topics, such as the Ricci flow and the Yamabe flow. The solvability of the Cauchy problem for these equations, the study of the blow up profile of solutions, the nonradial structure solutions, and the uniqueness of solutions, are among the problems which will be investigated in this specific area. The still open question of the uniqueness of sign-solutions of the porous medium equation, in connection with the continuity question for a class of linear singular parabolic equations falls under the proposed major goals to be studied. The non-linear equations to be studied under this proposal form the basic concepts of many applications which deem to be important to technology and the society at large. The purification of materials, from chemicals to petroleum and even water, is often achieved by diffusion through filters. The purification filters are the porous media described in the proposal. Thin film dynamics and the Van der Waals forces operating between thin layers are described by singular quasilinear equations of ultra-fast d iffusion. The dynamics of population growth, polymer chain growth, including cross linking and high rate growth of biomolecules, are also non-linear phenomena amiable to our basic studies. The interesting problem of the expanding universe and other cosmological phenomena seem to be goverened by nonlinear dynamics, which in certain cases are applications of the more complex problems described here.