This project is devoted to a mathematical study of some nonlinear partial differential equations (PDE) arising from physical and biological sciences. The principal investigator plans to address a singular activator-inhibitor system modeling the regeneration phenomena in biology. The project seeks to understand the mechanism of pattern formation, so the long-time behavior, stability, and bifurcation of the solutions will be analyzed. The principal investigator will also study various free boundary problems with applications to thin film dynamics, liquid crystal configurations, and crystal solidification processes. Here the focus is on the structure and dynamics of the singular set and the existence, uniqueness, and regularity of the free boundaries involved. Due to the inherent singularity and nonlinearity of the problems, it is necessary to use analytical tools from nonlinear elliptic and parabolic PDE theory, harmonic analysis, the calculus of variation, geometric measure theory, and geometric analysis.
Partial differential equations have been broadly used to model a wide variety of phenomena in physics, biology, chemistry, and finance. Many mathematically challenging problems arise when the equations involve singularities. Examples of singularities include dry spots of thin films, sharp corners in liquid/solid interfaces, and unbounded physical quantities. This project will develop novel ideas and methods to study singularities in the relevant nonlinear partial differential equations and hence yield new understanding of the underlying physical and biological processes. The project will also have an impact on the development of human resources: it will supply teaching materials in graduate courses and provide advanced training for undergraduate and graduate students, thereby allowing them to participate in various aspects of the project.
