反應擴散方程的非平面傳動波

Abstract

這篇論文主要處理的問題是反應擴散方程的傳動波ut=△u+uzz-f(u)，其中(x, z)∈Rn+1是空間變數。假定f(u)是一個双雙穩定的非線性項，我們分別考慮平衡的狀態及非平衡的狀態。在平衡的狀態下，我們將描述多種連接兩個平衡點的傳動波，這些傳動波各有各的形狀。在非平衡的狀態下，我們將傳動波解限制為柱狀對稱的形式，接著證明這樣的解在n≥2時其形狀會近似於拋物面，而在n=1時會近似於超餘弦函數。除此之外，我們也將證明單穩傳動波的存在性。本篇論文的主要參考文獻的作者有以下幾位：Y. Morita、H. Ninomiya、X.F. Chen、J-S Guo、F. Hamel及J-M Roquejoffre。

We are dealing with traveling wave solutions of a reaction-diffusion equation ut=△u+uzz-f(u), where (x, z)=(x1,---, xn, z) ∈Rn+1 is the space variable and △ is the Laplacian in Rn. Assume that f(u) is a bistable nonlinear, then we consider the balanced case and unbalanced case respectively. In the preceding case, we describe some types of traveling waves connecting two stable equilibria. In the case of latter, we want to find out the bistable-type traveling waves with the interfaces other than plane. If the solution is restricted to be cylindrically symmetric, then we can show that the interface is asymptotically a paraboloid as n≥2 and a hyperbolic cosine curve as n=1. Besides, we prove the existence of the monostable-type traveling waves. The main references of this thesis are Y. Morita, H. Ninomiya, X.F. Chen, J-S GUO, F. Hamel and J-M Roquejoffre.