Special Solutions of Some Semilinear Parabolic Equations and Their Discrete Analogue
| Autor: | Chin-Chin Wu, 吳菁菁 |
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| Rok vydání: | 2007 |
| Druh dokumentu: | 學位論文 ; thesis |
| Popis: | 96 In this thesis, we study special solutions of some semilinear parabolic equations and their discrete analogue. In the first part, we study the existence, uniqueness, and stability of traveling waves for a system of ordinary differential equations with bistable nonlinearity in discrete periodic media. This system arises from a spatial discrete version of some semilinear parabolic equations with periodic nonlinearity. The main tools to derive the uniqueness and asymptotic stability are comparison principle, spectrum analysis of the linearized operator around a steady state, and the construction of suitable super/subsolutions. To derive the existence of traveling wave, we first convert the system to an integral equation. Then we establish the existence traveling wave for this system of ordinary differential equations. In the second part, we study the solution of initial boundary value problem for the heat equation with a strong absorption term. It is well-known that the solution develops a dead-core in finite time for a large class of initial data. It is also known that the exact dead-core rate is faster than the corresponding self-similar rate. By using the idea of matching, we formally derive the exact dead-core rates under a dynamical theory assumption. Moreover, we also construct some special solutions for the corresponding Cauchy problem satisfying this dynamical theory assumption. These solutions provide some examples with certain given polynomial rates. |
| Databáze: | Networked Digital Library of Theses & Dissertations |
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