幾個非線性方程解的結構
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2004
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在本篇論文，我們首先研究一個半線性常微分方程的初值問題。這個問題與一個半線性拋物型方程的爆破解之漸近行為有很密切的關係。在某些適當的條件下，我們刻畫此常微分方程解的結構，並由此導出該初值問題的整體解之唯一性。
接下來，我們研究一個具有suplinear反應項的非線性拋物型方程。藉由研究該方程的backward之自我相似解，我們構造出有限個僅在單一點爆破的自我相似解，並且這些解所具有之極值點的個數是互不相同。
最後，我們研究一個三階常微分方程的邊界值問題(BVP)，這個問題起源於：在飽和介質中，垂直放置一塊不具有滲透性的加熱平板，並考慮具有穩定自由對流的自我相似解。我們考慮此三階常微分方程之初值問題的所有解的結構。首先，我們將解分成六類。然後，經過適當的變換，將此三階常微分方程轉換為一個二階常微分方程，再利用比較原理，我們可以得到所有解的初步結構。這也就回答了Belhachmi，Brighi 與 Taous在2001年所提出的一些open問題。為了進一步區分這些解，我們引進一個新的變換，該變換將此三階常微分方程轉換為兩個一階常微分方程所組成之方程組。然後藉由相平面的分析，我們得到解的更精細之結構。
In this thesis, we first study an initial value problem for a semilinear ordinary differential equation. This problem is closely related to the blow-up behaviour of a semilinear parabolic equation. Under some restriction, we characterize the structure of solutions and derive the uniqueness of positive global solution of this initial value problem. Next, we study a nonlinear parabolic equation with a superlinear reaction term. By studying the backward self-similar solutions for this equation, we construct a finite number of self-similar single-point blow-up patterns with different oscillations. Finally, we study a boundary value problem for a third order differential equation which arises in the study of self-similar solutions of the steady free convection problem for a vertical heated impermeable flat plate embedded in a porous medium. We consider the structure of solutions of the initial value problem for this third order differential equation. First, we classify the solutions into 6 different types. Then, by transforming the third order equation into a second order equation, with the help of some comparison principle we are able to derive the structure of solutions. This answers some of the open questions proposed by Belhachmi, Brighi, and Taous in 2001. To obtain a further distinctions of the solution structure, we introduce a new change of variables to transform the third order equation into a system of two first order equations. Then by the phase plane analysis we can obtain more information on the structure of solutions.
In this thesis, we first study an initial value problem for a semilinear ordinary differential equation. This problem is closely related to the blow-up behaviour of a semilinear parabolic equation. Under some restriction, we characterize the structure of solutions and derive the uniqueness of positive global solution of this initial value problem. Next, we study a nonlinear parabolic equation with a superlinear reaction term. By studying the backward self-similar solutions for this equation, we construct a finite number of self-similar single-point blow-up patterns with different oscillations. Finally, we study a boundary value problem for a third order differential equation which arises in the study of self-similar solutions of the steady free convection problem for a vertical heated impermeable flat plate embedded in a porous medium. We consider the structure of solutions of the initial value problem for this third order differential equation. First, we classify the solutions into 6 different types. Then, by transforming the third order equation into a second order equation, with the help of some comparison principle we are able to derive the structure of solutions. This answers some of the open questions proposed by Belhachmi, Brighi, and Taous in 2001. To obtain a further distinctions of the solution structure, we introduce a new change of variables to transform the third order equation into a system of two first order equations. Then by the phase plane analysis we can obtain more information on the structure of solutions.
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拋物型方程, 爆破解, 自我相似解, 相平面, parabolic equation, blow-up solution, self-similar solution, phase plane