穩態中的不穩定狀態以偏微分方程為出發之研究與探討

Abstract

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]。

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].