三维趋化模型初边值问题的全局分析

谢卫军, 张映辉

(1. 湖南民族职业学院 初等教育系, 湖南 岳阳414000; 2. 湖南理工学院 数学学院, 湖南 岳阳 414006)

三维趋化模型初边值问题的全局分析

谢卫军1, 张映辉2

(1. 湖南民族职业学院 初等教育系, 湖南 岳阳414000; 2. 湖南理工学院 数学学院, 湖南 岳阳 414006)

建立了三维趋化模型初边值问题强解的全局存在性和大时间行为. 具体地说, 得到了全局H2强解的存在性和指数稳定性. 证明方法是基于精细的能量估计.

趋化模型; 强解; 存在性; 指数稳定性

如果用变量(x,t)取代(τ,ξ), 则(8)恰好是(1).

对于系统(1)的一维情形, 文[3]和文[4]分别考虑了相应的初边值问题和柯西问题. 文[3]中, 作者考虑了零Dirichlet边值问题. 当充分小时, 他们得到了光滑解的全局存在性. 文[4]中, 作者证明了大初值问题光滑解的整体存在性. 在文[5]中, 我们得到了全局大解的最优收敛率. 至于系统(1), 文[6～7]分别证明了系统(1)柯西问题和初边值问题全局小H3强解的存在性和渐近行为. 最近, 我们在文[8]中证明了柯西问题全局小H2强解的存在性和最优衰减率. 有关其他的结果, 读者可参阅[9～14]以及其中的参考文献.

本文旨在研究初边值问题(1)～(3)H2强解的全局存在性和渐近行为. 对于充分小初始值, 我们得到了三维趋化模型初边值问题(1)～(3)全局小H2强解的存在性和指数稳定性.

本文的主要结果为:

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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

国家自然科学基金项目 (11301172); 湖南省教育厅优秀青年项目(14B077); 湖南省教育厅一般项目(14C0536)

谢卫军(1980− ), 男, 湖南常宁人, 湖南民族职业学院初等教育系讲师. 主要研究方向: 偏微分方程