Analysis on Singularities of Solutions to Nonlinear Partial Differential Equations

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The purpose of this research is to study the behavior of solutions to nonlinear partial differential equations by analyzing singularities of finite-time blowup solutions. Especially, we focus on two types of advection-diffusion equations (parabolic-elliptic system, parabolic system) and obtained the following results.1. The possibility of finite-time blowup of nonradial solutions to the parabolic-elliptic system.2. Finite-time blowup solutions to the parabolic system have δ-function singularities at isolated blow-up points in two-dimensional domains3. The structure of self-similar solutions to the parabolic system.4. The structure of steady states to the parabolic system.