穩態中的不穩定狀態以偏微分方程為出發之研究與探討

A Survey of Instability of Steady States

蕭煜修(Yu-Hsiu Hsiao)

指導教授 ： 王藹農

國立臺灣大學/理學院/數學研究所/碩士(2014年)

https://doi.org/10.6342/NTU.2014.02261

摘要

在這篇論文中,我們主要討論拋物方程與雙曲方程穩態解的情形。 我們會為讀者準備充分的先備知識,由淺入深地從基本的定義開始介 紹,最終會接到我們的主題 -由橢圓方程 Lφ = f(x,φ),x ∈ Rn 出發,其 中 φ 是時間獨立的解,在某些假定的條件之下,我們將可以由穩態的解 中導出不穩定狀態的結果,為了完成我們的工作,我們主要的參考文獻為 Manoussos Grillakis, Jalal Shatah, 以及 Walter Strauss 合力完成的 [2] 與 Paschalis Karageorgis, Walter A Strauss 共同完成的 [3],基本知識的準備 我們主要參考 Lawrence C. Evans 的著作 [1] 與 Walter A. Strauss 的著作 [4]。

關鍵字

非線性熱方程 ； 非線性波方程 ； 穩態 ； 時間獨立

並列摘要

We consider the steady states solutions of parabolic and hyperbolic equa- tions such as ∂tu − ∆u = f(x, u) and ∂ttu − ∆u = f(x, u). Steady state which means a system that has numbers of properties that are unchanged in time. For instance, property p of the steady state system has zero partial derivative with respect to time : ∂p = 0.
∂t
In this thesis we will give a proof about the instability results about the solutions of a general elliptic equation of the form Lφ = f (x, φ),x ∈ Rn ,where L is a linear,second-order elliptic differential operator whose coefficients are smooth and bounded. φ is the time-independent solution of Lu = f (x, u),x ∈ Rn. To complete our work, we mainly consult paper[2] and [3].Also for some basic preliminaries we consult text books[1] and [4].

並列關鍵字

Nonlinear heat equation ； Nonlinear wave equation ； Steady states ； Instability

參考文獻

[1] Lawrence Evans. Partial differential equations. 1998.

[2] Manoussos Grillakis, Jalal Shatah, and Walter Strauss. Stability theory of solitary waves in the presence of symmetry, i. Journal of Functional Analysis, 74(1):160–197, 1987.

[3] Paschalis Karageorgis and Walter A Strauss. Instability of steady states for nonlinear wave and heat equations. Journal of Differential Equations, 241(1):184–205, 2007.

[4] Walter A Strauss. Partial differential equations. an introduction. New York, 1992.

[5] Grozdena Todorova and Borislav Yordanov. Critical exponent for a non- linear wave equation with damping. Journal of Differential Equations, 174(2):464–489, 2001.