Analysis on Singularities of Solutions to Nonlinear Partial Differential Equations

本研究旨在通过分析有限时间爆破解的奇性来研究非线性偏微分方程解的性态。特别地，我们研究了两类对流扩散方程(抛物型椭圆型方程组和抛物型方程组)，得到了以下结果。抛物-椭圆型方程组非径向解的有限时间爆破的可能性。抛物型方程组的有限时间爆破解在二维域中孤立爆破点具有δ函数奇性。抛物型方程组自相似解的结构。抛物型系统的定态结构。

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.