三維趨化模型初邊值問題的全局分析

謝衛(wèi)軍, 張映輝

(1. 湖南民族職業(yè)學院 初等教育系, 湖南 岳陽414000; 2. 湖南理工學院 數學學院, 湖南 岳陽 414006)

三維趨化模型初邊值問題的全局分析

謝衛(wèi)軍1, 張映輝2

(1. 湖南民族職業(yè)學院 初等教育系, 湖南 岳陽414000; 2. 湖南理工學院 數學學院, 湖南 岳陽 414006)

建立了三維趨化模型初邊值問題強解的全局存在性和大時間行為. 具體地說, 得到了全局H2強解的存在性和指數穩(wěn)定性. 證明方法是基于精細的能量估計.

趨化模型; 強解; 存在性; 指數穩(wěn)定性

如果用變量(x,t)取代(τ,ξ), 則(8)恰好是(1).

對于系統(tǒng)(1)的一維情形, 文[3]和文[4]分別考慮了相應的初邊值問題和柯西問題. 文[3]中, 作者考慮了零Dirichlet邊值問題. 當充分小時, 他們得到了光滑解的全局存在性. 文[4]中, 作者證明了大初值問題光滑解的整體存在性. 在文[5]中, 我們得到了全局大解的最優(yōu)收斂率. 至于系統(tǒng)(1), 文[6～7]分別證明了系統(tǒng)(1)柯西問題和初邊值問題全局小H3強解的存在性和漸近行為. 最近, 我們在文[8]中證明了柯西問題全局小H2強解的存在性和最優(yōu)衰減率. 有關其他的結果, 讀者可參閱[9～14]以及其中的參考文獻.

本文旨在研究初邊值問題(1)～(3)H2強解的全局存在性和漸近行為. 對于充分小初始值, 我們得到了三維趨化模型初邊值問題(1)～(3)全局小H2強解的存在性和指數穩(wěn)定性.

本文的主要結果為:

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Global Analysis of Solutions to Initial Boundary Value Problems for the 3D Chemotaxis Model

XIE Wei-jun1, ZHANG Ying-hui2
(1. Department of Primary Education Hunan, National Vocational College, Yueyang 414000, China; 2. College of Mathemaics, Hunan Institute of Science and Technology, Yueyang 414006, China)

This paper establishes global existence and asymptotic behavior of strong solutions to the initial boundary value problem for the 3D chemotaxis model. More precisely, existence and exponential stability of globalH2strong solutions is obtained. The proof is based on delicate energy estimates.

chemotaxis model; strong solution; existence; exponential stability

O175.2

A

1672-5298(2015)02-0010-03

2015-04-12

國家自然科學基金項目 (11301172); 湖南省教育廳優(yōu)秀青年項目(14B077); 湖南省教育廳一般項目(14C0536)

謝衛(wèi)軍(1980? ), 男, 湖南常寧人, 湖南民族職業(yè)學院初等教育系講師. 主要研究方向: 偏微分方程