拋物型問題的奇異點研究

Abstract

在本論文中，我們要討論從二個拋物型方程得到的二種不同類型的奇異點問題。本論文分為二個部份， 在第一部份中，我們考慮具有快速擴散項與強吸收非線性項之方程的殆核問題。首先，我們證明解殆核的速度是非自我相似的。接著，在考慮重新縮放的解與殆核最終在單點發生的狀態下，我們得到一些更精確的估計。 在第二部份中，我們探討一個由複數取值的熱方程得到的柯西問題，而其中的非線性項是倒數型的。首先，我們提供了一些解的全局存在性與消失性的判斷準則。接下來，我們證明當初始值為漸近常數時，解是否會在無窮遠處消失或是在任意的時間內全局存在，均依賴於初始值的漸近極限值。

In this thesis, we study two different singularities arising from two parabolic problems. This thesis is divided into two parts. In the first part, we consider the dead-core problem for the fast diffusion equation with a strong absorption. First, we show that the temporal rate of formation of the dead-core is not self-similar. Then we obtain some precise estimates on rescaled solutions and on the single-point final dead-core profile. In the second part, we study the Cauchy problem for a parabolic system which is derived from a complex-valued heat equation with an inverse nonlinearity. We first provide some criteria for the global existence and quenching of solutions. Then we show that, for the initial data which are asymptotically constants, the solution either quenches at space infinity or exists globally in time depending on the asymptotic limits.