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研究生:陳德軒
研究生(外文):De-Syuan, Chen
論文名稱:具有非線性邊界條件之多孔介質方程解的漸近行為
論文名稱(外文):Asymptotic behavior of solutions to a porous medium equation with a nonlinear boundary condition
指導教授:鄭博仁鄭博仁引用關係
指導教授(外文):Po-Jen, Cheng
學位類別:碩士
校院名稱:國立嘉義大學
系所名稱:應用數學系研究所
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
畢業學年度:96
語文別:英文
中文關鍵詞:爆炸熄滅多孔介質方程穩定解
外文關鍵詞:blow-upquenchingporous medium equationstationary solution
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在這篇論文中,我們研究一個具有非線性邊界條件以及初始條件的多孔介質方程,文章的主要目的是研究多孔介質方程的爆炸與熄滅現象。藉由針對諾伊曼邊界條件的比較定理,以及問題的初始條件,我們定義出發生爆炸現象以及熄滅現象的充分條件。我們更進ㄧ步的討論相對的穩定問題,並且證實了穩定解的存在性、唯一性以及不穩定性。
In this thesis, we study a porous medium equation with a nonlinear boundary condition and an initial condition. Our main purpose is to study blow-up and quenching phenomenon. Using the comparison principle for the nonlinear Neumann boundary condition and the initial condition, we give the criteria for blow-up and quenching by initial datum. Furthermore, we also study the stationary problem, establish the existence, uniqueness and instability of the stationary solution.
Chinese Abstract ………………………………………………… Ⅰ
Abstract …………………………………………………………… Ⅱ
Acknowledgement …………………………………………………… Ⅲ
Table of Contents ………………………………………………… Ⅳ

Chapter 1 Introduction ………………………………………… 1

Chapter 2 Blow-up and Quenching …………………………… 4

2.1 Introduction ………………………………………………… 4
2.2 The comparison principle for the nonlinear
Neumann boundary condition ……………………………… 5
2.3 Criteria for blow-up at finite time ………………… 8
2.4 Criteria for quenching at finite time ……………… 12

Chapter 3 Porous Medium Equation with
Steady State Solution …………………………… 15
3.1 Introduction ……………………………………………… 15
3.2 The stationary problem ………………………………… 16
3.3 Existence and uniqueness of stationary solutions… 18
3.4 Instability of stationary solutions ………………… 21

Bibliography ……………………………………………………… 25

A. Acker and W. Walter, The quenching problem for nonlinear parabolic differential equations, Lecture Notes in Math. 564, Springer, Berlin, 1976, 1-12.

Y. Chen and C. Xie, Blow-up for a porous medium equation with a localized source, Appl. Math. Comput. 159 (2004), No. 1, 79-93.

K. Deng and C.L. Zhao, Blow-up versus quenching, Commun. Appl. Anal. 7 (2003), No. 1, 87-100.

K. Deng and C.L. Zhao, Instability of solutions of a semilinear heat equation with a Neumann boundary condition, Quart. Appl. Math. 63 (2005), No. 1, 13-19.

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ, 1964.

H. Kawarada, On solutions of initial-boundary problem for u_t=u_{xx}+1/(1-u), Publ. Res. Inst. Math. Sci. 10 (1975), No. 3, 729-736.

Y.W. Qi, The existence of moving boundary solution of a porous media equation with a source term, Adv. Math. Sci. Appl. 6 (1996), No. 1, 197-215.

J. L. Vazquez, The interfaces of one-dimensional flows in porous media, Trans. Amer. Math. Soc. 285 (1984), No. 2, 717-737.

J. L. Vazquez, The Porous Medium Equation: Mathematical Theory, Oxford University Press, 2007.

Y. Zhi and C. Mu, The quenching behavior of a nonlinear parabolic equation with nonlinear boundary outflux, Appl. Math. Comput. 184 (2007), No. 2, 624-630.

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