Qualitative Studies of Some Partial Differential Equations and Systems

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PI: Changfeng Gui, University of ConnecticutDMS-0140604Abstract:The PI will work on two projects on qualitative properties of partial differential equations. In the first project, he will study phase transitions modeled by the Allen-Cahnequation and its generalized vector equations, which are the gradient flows of the Allen-Cahn energy with either double-well potentials or multiple-well potentials of equal depths. The objectives are to determine the existence, uniqueness and stability of some special configurations, and then to further study the fine structure near the interfaces or nearthe triple junction and the dynamics of these sharp interfaces andtriple junctions. One of the immediate goals is to classify the optimalconfigurations of anti-phases. This is indeed to solve a conjecture ofDe Giorgi which concerns the one dimensional symmetry ofcertain entire solutions of the Euler-Lagrange equations of theAllen-Cahn energy. The De Giorgi conjecture is also related to the studyof minimal surfaces, and is still open in dimensions bigger than three.In the second project the PI will study the Gierier-Meinhardtsystems of equations arising in biological pattern formations and chemicalreactions. In particular, the proposor will try to show mathematicallythe existence of some special concentration patterns and to understandtheir stabilities.Diffusion is a very common phenomenon, and it generates verycomplex structures when several different substances are involved.Phase transitions, pattern formations are well observed in materialsciences, in biological and chemical reactions, and they are closelyrelated to diffusions. Special nonlinear partial differential equationsand systems are used to model these phenomena mathematically. Theproposor hope to develop analytic techniques to obtain goodqualitative properties for solutions of these equations, which will leadto better numerical methods for simulations. The study will help tounderstand the finer structures and long time behaviors of phasetransitions, pattern formations, and other similar phenomena.