Qualitative Studies of Some Partial Differential Equations and Systems

University Of Connecticut, Storrs CT

Investigators

Abstract

PI: Changfeng Gui, University of Connecticut DMS-0140604 Abstract: 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-Cahn equation 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 near the triple junction and the dynamics of these sharp interfaces and triple junctions. One of the immediate goals is to classify the optimal configurations of anti-phases. This is indeed to solve a conjecture of De Giorgi which concerns the one dimensional symmetry of certain entire solutions of the Euler-Lagrange equations of the Allen-Cahn energy. The De Giorgi conjecture is also related to the study of minimal surfaces, and is still open in dimensions bigger than three. In the second project the PI will study the Gierier-Meinhardt systems of equations arising in biological pattern formations and chemical reactions. In particular, the proposor will try to show mathematically the existence of some special concentration patterns and to understand their stabilities. Diffusion is a very common phenomenon, and it generates very complex structures when several different substances are involved. Phase transitions, pattern formations are well observed in material sciences, in biological and chemical reactions, and they are closely related to diffusions. Special nonlinear partial differential equations and systems are used to model these phenomena mathematically. The proposor hope to develop analytic techniques to obtain good qualitative properties for solutions of these equations, which will lead to better numerical methods for simulations. The study will help to understand the finer structures and long time behaviors of phase transitions, pattern formations, and other similar phenomena.