Mathematical Sciences: Studies on Nonlinear Partial Differential Equations

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9418779 Zou This award supports mathematical research on problems arising in two areas of nonlinear partial differential equations. The first is related to the analysis of the asymptotic behavior and classification of positive solutions of semilinear equations in n-dimensional space, focusing on the Lane-Emden equation. The problem has been studied exhaustively in various special cases. Extensions and generalizations of this work will also be carried out. Uniqueness of positive solutions, especially in the case of radial solutions will also be taken up. A new direction of research concerns existence and non-existence questions for semilinear systems such as the Fujita system. More applied studies will consider the thermistor problem, analyzing electrical resistors whose resistance increases rapidly with increasing temperature. Finally, work will be done on the existence problem for ground states for quasilinear elliptic quations and the Dirichlet problem for the Monge-Amper Equation Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. ***