Mathematical Sciences: Semiliner Partial Different EquationsCurve Shortening in Minkowski Geometry and Branching Processes in Probability

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Li 9225145 This project concentrates on three areas of research in mathematical analysis. The general theme is that of determining existence, regularity, local and global behavior and symmetry properties of solutions of nonlinear equations and systems. In particular, work on semilinear elliptic partial differential equations will continue, examining the Lane-Emden equation and Matukuma equation of astrophysics. This includes a complete classification of positive solutions, and to determine whether all solutions are radial with limiting values of zero or infinity. A second line of investigation considers anisotropic curve shortening. This has a more geometric flavor in which embedded plane curves evolve according to a rate proportional to the normal vector. The present work focuses on a generalized setting of curves on manifolds in a Minkowski geometry. Efforts will be made to determine when such flows can shrink to a point when standing symmetry hypotheses are not present. The ideas represented by this work can be viewed as models for the evolution of phase boundaries such as the boundaries of growing crystals. The third direction this works takes involves measure-valued branching processes, including the model on the evolution caused by point catalysts. *** The work has its roots in stochastic partial differential equations arising in the study of spatially distributed birth and death processes. A particular goal is to analyze nonequilibrium reactions in the presence of multiple catalysts.