Asymptotics, Existence, Symmetry and Uniqueness in NonlinearPartial Differental Equations

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Abstract Zou This proposal is concerned with problems of existence, qualitative properties (e.g., asymptotics, symmetry, etc.) and uniqueness of solutions for some model nonlinear elliptic and parabolic equations as well as systems. More specifically, efforts will be devoted to investigations of the asymptotic behavior and symmetry of positive solutions for the Lane - Emden equation and general nonlinear elliptic equations and systems, the existence and non-existence for positive solutions of the Lane - Emden system, the scalar field system and a semilinear Hamiltonian system, the asymptotics of solutions of Ginzburg-Landau equations, the existence and regularity of solutions to a thermistor problem and the uniqueness of positive solutions of semilinear elliptic equations with emphasis on the impact of the geometry of the domain. Systematic methods and approaches for general problems will be developed by studying these model equations. Tools for this project include a variational approach, the moving plane method, the topological fixed point theory and degree theory, and, asymptotic and a priori estimates are essential. Partial differential equations form a basis for mathematical modeling of the physical world. Our model problems in this proposal arise from the study of various nonlinear phenomena, including pattern formation and population evolution (mathematical biology), the stellar structure in astrophysics and superconductivity (physics), the thermistor problem (industrial mathematics) and the Yamaba problem (differential geometry). It is, in studying classical partial differential equations, fundamental and essential to provide qualitative and quantitative information about the solutions, which lies in the core of mathematical analysis. This may include answers to questions about uniqueness, smoothness, shape and asymptotic behavior of solutions. The proposed investigations in this proposal are to provide such studies for our model problems. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations.