臺灣博碩士論文加值系統

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

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