﻿ 拟单调中立型反应扩散方程行波解的唯一性 Uniqueness of Traveling Wave Solutions for a Quasi-Monotone Reaction-Diffusion Equation with Neutral Type

Pure Mathematics
Vol.07 No.04(2017), Article ID:21402,12 pages
10.12677/PM.2017.74041

Uniqueness of Traveling Wave Solutions for a Quasi-Monotone Reaction-Diffusion Equation with Neutral Type

Yubin Liu

College of Mathematics and Data Science, Huizhou University, Huizhou Guangdong

Received: Jun. 29th, 2017; accepted: Jul. 13th, 2017; published: Jul. 19th, 2017

ABSTRACT

In present paper, we focus on the uniqueness of traveling wave solutions for a quasi-monotone reaction-diffusion equation with neutral type. By using the Ikehara’s Theorem, we firstly establish the asymptotic exponent properties of monotone traveling wave solution with speed for the reaction-diffusion equation with a infinite number of delays, which is transformed from the neutral equation by a linear variable transform, and then the uniqueness (up to translation) of monotone traveling wave solution with speed for the transformed equation. Finally, we obtain the uniqueness (up to translation) of monotone traveling wave solution with speed for the neutral equation by using the relation between solutions of the neutral equation and of the transformed equation.

Keywords:Reaction-Diffusion Equation with Neutral Type, Traveling Wave Solution, Quasi-Monotone Reaction, Uniqueness

1. 引言

(1.1)

2. 预备知识

(H1) 对任意；对任意

(H2) 对任意关于单调增加，且存在，使的对任意，有

(H3)，且对任意的，有

(H4) 存在，使得，且对任意，有

(H5) 存在，使得对任意的，存在唯一的，使得，且对任意，对任意

(H6)

(1)

(2) 对，当时，有

(3) 对任意，存在，使得，且对任意的

(1) 对方程(1.1)的行波解存在，在上单调递增，且满足

,

(2) 对，方程(1.1)不存在行波解。

(2.1)

(1) 对方程(2.1)的行波解存在，在上单调递增，且满足

,

(2) 对方程(2.1)不存在行波解。

3. 行波解的唯一性

,

(3.1)

(3.2)

(3.3)

，所以存在，使得对所有的，有，从而对所有的，有

(3.4)

，由于收敛，故有

(3.5)

，有

，取；记，则。取，则。定义，则

(3.6)

.

Uniqueness of Traveling Wave Solutions for a Quasi-Monotone Reaction-Diffusion Equation with Neutral Type[J]. 理论数学, 2017, 07(04): 310-321. http://dx.doi.org/10.12677/PM.2017.74041

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