Free Boundary Problems and Reaction-Diffusion Systems

University Of Pittsburgh, Pittsburgh PA

Investigators

Abstract

DMS Award Abstract Award #: 0203991 PI: Chen, Xinfu Institution: University of Pittsburgh Program: Applied Mathematics Program Manager: Catherine Mavriplis Title: FREE BOUNDARY PROBLEMS AND REACTION-DIFFUSION SYSTEMS This project addresses interfacial phenomena that are commonplace in nature. There are two goals in this project. One is to develop an applicable model describing motion of soluble substances in a liquid environment. The other is to use existing reaction--diffusion systems and free boundary models to study certain interfacial phenomena that are important in industrial applications. When chemical reactions are concerned, existing models to be used are reaction--diffusion systems such as the activator--inhibitor, FitzHugh--Nagumo, Belousov-Zhabotinsky, Gray--Scott, and Gierer--Meinhardt. Most of the focus here will be on slow diffusion--fast reaction equations and their singular limits, namely, free boundary problems. A large part of the investigation will be to find solutions related to interfacial phenomena, such as traveling spots, spikes, target patterns, checkerboard patterns, rotating spiral waves, and self--replicating behavior. The underlying investigation for patterns in free boundary problems and reaction--diffusion systems will enable people to predict and to control the seemingly complicated yet natural interfacial dynamics. It can also develop mathematical theories and ideas. As a particular example, certain evolution equations that do not possess uniqueness and thus classically are regarded as ill--posed, will be carefully investigated and regarded as good models since they reflect the nature in which an unobservable local microscopic change may affect the whole observable macroscopic outcome. Such a research will be guided by the current industrial needs. In chemistry, biology, and many applied sciences, it is common that chemical substances move around in a liquid environment, due to dissolution or chemical reaction. Quite often circular shapes are stable and circular spots move and interact each other like elastic objects, with their own special ``momentum laws". One purpose of the project is to develop, at least for soluble solids in a liquid medium, a model enabling one to calculate the speed of a traveling spot and the reflection principle---the relationship between the incoming and outgoing angles when a traveling spot bounces off another spot or the physical boundary of a container. The model developed for the motion of spots is expected to find useful industrial applications in many areas such as water contamination control, chemical testing, chemical processing, etc. This project will involve graduate students and stimulate their interest in mathematical problems of practical importance. Date: June 14, 2002