Collaborative Research: Nonlinear PDE's and Integro-Differential Equations in the Complex Plane

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Abstract: 0406193 O Costin, Rutgers UniversityCollaborative research on nonlinear PDEs and integro-differential equations The proposal develops and applies new constructive methods in the study of solutions of nonlinear partial differential equations, with focus on existence and uniqueness, regularity and spontaneous formation of singularities. Recent advances in Borel-Laplace regularization andsummability are used to find global properties of solutions ofPDEs. There is a wide range of problems where these questions areessential, and a number of concrete examples including thin filmequations, Kuramoto-Sivashinski, KdV and mKdV will be analyzed.Natural phenomena in physics, chemistry, material sciences and in manyother disciplines are most often described by differential equations;complex phenomena are frequently modeled by partial differentialequations. The present project covers a wide range of such equations,the global properties of their solutions and their possible blowup. This resarch is a piece of an unfolding enterprise to advance atheory of singular behavior, of substantial importance in mathematicsand its applications.