一些非線性拋物型問題的爆破

Blow-up for some nonlinear parabolic problems

符聖珍(Sheng-Chen Fu)

指導教授 ： 郭忠勝

國立臺灣師範大學/理學院/數學系/博士(2002年)

摘要

在本篇論文，我們首先討論一個半線性熱方程解的爆破行為，其中我們所考慮的邊界條件是非線性的。在某種假設下，我們得到解只會在邊界爆破。接著，利用Giga-Kohn轉換，我們得到解趨近爆破時間的漸進行為。此外，我們得到這種爆破是徹底的(complete)。
接著，我們討論一個非線性反應擴散系統解的爆破行為。我們主要的目的是要瞭解反應項和吸收項對爆破性質的影響。在某種假設下，我們得到爆破的充分必要條件，爆破速度的上下界估計，及爆破集合。

關鍵字

爆破 ； 半線性熱方程 ； 反應擴散系統

並列摘要

In this thesis, we first study the blow-up
behaviors of solutions of a semilinear heat
equation with a nonlinear boundary condition.
Under certain conditions, we prove that the
blow-up point occurs only at the boundary.
Then, by applying the well-known method of
Giga-Kohn, we derive the time asymptotic of
solutions near the blow-up time. In addition,
we prove that the blow-up is complete.
Next, we study the blow-up behavior for
a semilinear reaction-diffusion
system coupled in both equations and boundary conditions.
The main purpose is to understand how the reaction terms
and the absorption terms affect the blow-up properties.
We obtain a necessary and sufficient condition for blow-up,
derive the upper bound and lower bound for the blow-up rate,
and find the blow-up set under certain assumptions.

並列關鍵字

blow-up ； semilinear heat equation ； reaction-diffusion system

參考文獻

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