Mathematical Sciences: Further Development and Applications of the Method of Moving Planes

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Abstract 9623390 Li The PI will develop further the method of moving planes and study the following three problems: the a priori estimates, classification, and asymptotic analysis of solutions to some nonlinear differential equations; the prescribed Gaussian (or scalar) curvature problem; and the existence, uniqueness, and stability of travelling fronts. One focusing point of this project is to classify singular solutions or sequences of blow-up solutions in terms of when, where, how, and in what form do they occur. The method of moving planes is the main tool to be developed and used for this and other purposes. The analysis is particularly useful in the study of the last two problems where the PI has already obtained some interesting results. Many aspects of natural phenomena are related to each other by the natural laws governing them and very often these laws are described by differential equations. To understand these differential equations, it is necessary to classify all possible singular solutions or sequences of blow-up solutions. The PI proposes to study the prescribing curvature problem which involves some very challenging nonlinear partial differential equations. In fact, this type of equations also appear in many different fields of sciences. Also, singular solutions or sequences of blow-up solutions are the major barriers to the complete understanding of these equations. The PI will also study the existence, uniqueness, and stability of travelling fronts which describe the long term behavior of solutions to evolution type equations. He is mainly interested in problems arising from population dynamics and from flame propagation and ignition in combustion theory.