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研究生: 張雅菱
Ya-Ling Chang
論文名稱: 兩點自由邊界問題的向前自我相似解
Forward Self-similar Solution for A Two-point Free Boundary Problem
指導教授: 郭忠勝
Guo, Jong-Shenq
學位類別: 碩士
Master
系所名稱: 數學系
Department of Mathematics
論文出版年: 2003
畢業學年度: 91
語文別: 中文
論文頁數: 10
中文關鍵詞: 向前自我相似解兩點自由邊界問題
英文關鍵詞: Forward Self-similar Solution, A Two-point Free Boundary Problem
論文種類: 學術論文
相關次數: 點閱:163下載:4
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  • 在論文中,探討的是一個兩點自由邊界問題的自我相似解之存在性和唯一性。其中,將a(s)分成兩部分來探討:
    (A)若a(s)在無窮大處的一次微分大於0,則初始問題的解會完全地存在。
    (B)若a(s)在無窮大處的一次微分等於0,且存在一個有限的常數s0,使得a(s)在大於或等於s0時,它的二次微分都小於或等於0,則如果p足夠大時,初始問題的解是不會越過y軸,但是解一旦越過了y軸,它就會完全地存在。

    \section{Abstract}
    \hspace*{17pt}In this paper, we study the following two-point free boundary problem
    \begin{equation}\label{eq:(01)}
    \left\{
    \begin{array}{ll}
    u_t=(a(u_x))_x, \; -\xi_1(t) < x < \xi_2(t),\ t>0, & \\
    u_x(-\xi_1(t),t)=\tan\alpha_1, \;t>0,& \\
    u(-\xi_1(t),t)=\xi_1(t)\tan \beta_1,\; t>0, & \\
    u_x(\xi_2(t),t)=\tan\alpha_2, \; t>0, & \\
    u(\xi_2(t),t)=\xi_2(t)\tan \beta_2,\; t>0, & \\
    u(x, 0)=u_0(x),\ -\xi_{01}\le x\le \xi_{02},&\\
    \xi_1(0)=\xi_{01}, \; \xi_2(0)=\xi_{02}, &
    \end{array}
    \right.
    \end{equation}
    where $a\in C^2(\bf R)$ such that $a(0)=0$ and
    $a'>0$ in $\bf{R}$,
    $\beta_1, \beta_2 \in [0,\pi/2)$,
    $\alpha_1\in (-\beta_1,\pi/2)$,
    $\alpha_2\in(-\pi/2,\beta_2)$,
    $\xi_{01}$ and $\xi_{02}$ are positive constants,
    $u_0\in C^{1+\alpha}[-\xi_{01},\xi_{02}]$
    for some $\alpha\in(0,1)$,
    $u_0$ satisfies the compatibility conditions,
    and $u_0 > 0$ in $(-\xi_{01},\xi_{02})$.
    In this problem, $u,\xi_1,\xi_2$ are unknown functions to be determined.
    We are concerned with the existence of forward self-similar positive solutions of (\ref{eq:(01)}) in the form
    \begin{eqnarray}
    u(x,t)=\sqrt{2(t+1)}\varphi(\frac{x}{\sqrt{2(t+1)}})
    \end{eqnarray}
    for some smooth function $\varphi$.
    Then $u$ satisfies (\ref{eq:(01)}) if and only if $\varphi$ satisfies the following nonlinear free boundary problem (P):
    \begin{eqnarray}\label{eq:(02)}
    &&(a(\varphi'(y))'
    +y\varphi'(y) -\varphi(y) =0, -p <y < q, \\
    && \varphi(-p)=p\tan\beta_1, \varphi'(-p) =\tan\alpha_1,\\
    && \varphi(q)=q\tan\beta_2, \varphi'(q)=\tan\alpha_2
    \end{eqnarray}
    for some positive constants $p$ and $q$.
    The purpose of this paper is to study the existence and uniqueness of solution for the nonlinear free boundary problem (P).
    In this paper, we divide $a(s)$ into two case:
    $(\mbox{A})\;a^{'}(\infty)>0\label{eq:(04)}$ ;
    $(\mbox{B})\;a^{'}(\infty)=0\label{eq:(05)}$
    and there is a constant $s_{0}<\infty$
    such that $a^{''}(s)\leq 0$ for all $s\geq s_{0}$.
    In (A), the solution exists globally for all $p>0$.
    The special case "$a(s)=s$" of (A) has been proved in (1).
    For the case (B) it is more complicated.
    If $p$ is sufficiently large,
    then the solution of (IVP) cannot be extended over $y=0$.
    Moreover,
    the solution of (IVP) is global as long as it can be
    extended over $y=0$.
    By the comparison principle for nonlinear equations, we can
    obtain the existence and uniqueness of the self-similar
    solution.
    The special case "$a(s)=tan^{-1}s$" of (B) has been proved in (3).

    Contents 1. Introduction……………………………………………1 2. Preliminary……………………… ………………… 1 3. Existence and Uniqueness ………………………… 6 References…………………………………………………10

    (1) J.-S. Guo and Y. Kohsaka,
    Self-similar solutions of two-point free boundary Problem
    for the heat equation, Nonlinear Diffusive Systems and
    Related Topics, RIMS Kokyuroku 1258, Research Institute
    for Mathematical Sciences, Kyoto University, April, 2002,
    pp.94-107.
    (2) H.-H. Chern, J.-S. Guo and C.-P. Lo,
    The self-similar expanding curve for the curvature flow
    equation, Proc. Amer. Math. Soc.(to appear).
    (3) M. H. Protter and H. F. Weinberger, Maximum Principles
    in Differential Equations, Springer-Verlag, 1984.

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