Title

具一般非線性函數 Dirichlet-Neumann 邊界問題之分枝曲線分類與演化及其應用

Translated Titles

Classification and evolution of bifurcation curves for a Dirichlet-Neumann boundary value problem with general nonlinearity and its application

Authors

郭達昌

Key Words

Dirichlet-Neumann邊界值問題 ; 分枝曲線的演化 ; 分枝曲線的分類 ; 分枝曲線 ; 時間圖 ; Dirichlet-Neumann boundary value problem ; evolution of bifurcation curve ; classification of bifurcation curve ; bifurcation curve ; time map ; general nonlinearity

PublicationName

清華大學數學系學位論文

Volume or Term/Year and Month of Publication

2016年

Academic Degree Category

碩士

Advisor

王信華

Content Language

英文

Chinese Abstract

我們研究在Dirichlet-Neumann邊界條件下的正解分枝曲線的分類與演化,u''(x)+λf(u)=0, 0<x<1, u(0)=0, u'(1)=-c<0,這裡的λ>0,是分枝參數;而c>0,是演化參數。我們主要要證明函數f在適當的假設下,我們可以找到一個c₁>0,使得在(λ,‖u‖∞)平面上,我們有以下兩個性質。 (1)當0<c<c₁,分枝曲線為S型,而且在某些區間λ,會存在至少三個正解。 (2)當c≥c₁,分枝曲線為C型,而且在某些區間λ,會存在至少兩個正解。 我們的研究結果可以應用在一維的perturbed Gelfand equation,函數f(u)=exp((au)/(a+u))在a≥4.37。

English Abstract

We study the classification and evolution of bifurcation curves of positive solutions for the Dirichlet-Neumann boundary value problem u''(x)+λf(u)=0, 0<x<1, u(0)=0, u'(1)=-c<0, where λ>0 is a bifurcation parameter and c>0 is an evolution parameter. We mainly prove that, under some suitable assumptions on f, there exists c₁>0, such that, on the (λ,‖u‖∞)-plane, (i) when 0<c<c₁, the bifurcation curve is S-shaped and the problem has at least three positive solutions for some range of positive λ; (ii) when c≥c₁, the bifurcation curve is ⊂-shaped and the problem has at least two positive solutions for some range of positive λ. Our results can be applied to the one-dimensional perturbed Gelfand equation with f(u)=exp((au)/(a+u)) for a≥4.37.

Topic Category 基礎與應用科學 > 數學
理學院 > 數學系
Reference
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