Nonlinear Parabolic Equations: Free-Boundary Problems and Singular Behavior

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The major objectives of this proposal are concerned with the study of nonlinear parabolic equations related to degenerate and singular diffusion, in connection with more complex problems of differential geometry, including the Gauss curvature flow and the Ricci flowand with physical applications such as diffusion in porous media and thin liquid film dynamics.One specific area which will be investigated concerns with the regularity and geometry of interfaces in degenerate diffusion: this long term project which was initiated in 1997, has the objective of determining the connection between the geometry and regularity of the interfaces in degenerate diffusion. Another specific area of the main objectives concerns the study fast and super-diffusive nonlinear parabolic problems: this project has the objective of studying several new important phenomena related to the well-posedness and vanishing profile of solutions to singular diffusion equations arising in physical applications and differential geometry. In particular,it involves the study of the vanishing profile of the maximal solutions to the Ricci flow and the geometric implications of these results.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 super-fast diffusion. 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 governed by nonlinear dynamics, which in certain cases are applications of the more complex problems described here.