一类非线性Keller-Segel方程的局部零能控性

数学物理学报 ›› 2017, Vol. 37 ›› Issue (5): 834-845.

一类非线性Keller-Segel方程的局部零能控性

张亮, 杨国朋

武汉理工大学数学系武汉 430070

收稿日期:2016-11-18 修回日期:2017-04-16 出版日期:2017-10-26 发布日期:2017-10-26
作者简介:张亮,E-mail:zhangl@whut.edu.cn;杨国朋,E-mail:102213587@qq.com
基金资助:
国家自然科学基金（61573012）和湖北省自然科学基金（2014CFB337）

Null Controllability of a Nonlinear Keller-Segel Equation

Zhang Liang, Yang Guopeng

Department of Mathematics, Wuhan University of Technology, Wuhan 430070

Received:2016-11-18 Revised:2017-04-16 Online:2017-10-26 Published:2017-10-26
Supported by:
Supported by the NSFC(61573012) and the Hubei Natural Science Foundation (2014CFB337)

摘要/Abstract

摘要： 该文研究一类由抛物方程和椭圆方程耦合的非线性Keller-Segel方程的局部零能控性.该方程不仅具有非线性的drift-diffuion项，而且具有非线性的人口增长项.作者利用抛物-椭圆结构的非局部特性将方程组化为单个非线性抛物型方程并利用Kakutani不动点定理证明了局部零能控性的存在性.

关键词: 局部零能控性, Keller-Segel方程, 能观性估计

Abstract: In this paper, we study the local controllability for a nonlinear Keller-Segel equation coupled by an elliptic partial differential equation and a parabolic one, in which nonlinearity lies both on its drift-diffusion term and population growth. We prove the local null controllability by Kakutani's fixed point theorem. The method is established on the nonlocal structure of the elliptic-parabolic equation so that it can be treated as a single nonlinear parabolic equation.

Key words: Null controllability, Keller-Segel equation, Observability estimate

中图分类号:

O211.4

张亮, 杨国朋. 一类非线性Keller-Segel方程的局部零能控性[J]. 数学物理学报, 2017, 37(5): 834-845.

Zhang Liang, Yang Guopeng. Null Controllability of a Nonlinear Keller-Segel Equation[J]. Acta mathematica scientia,Series A, 2017, 37(5): 834-845.

参考文献

[1] Bonami A, Hilhorst D, Logak E, et al. Singular limit of a chemotaxis-growth model. Adv Differential Equations, 2001, 6:1173-1218
[2] Keller E F, Segel L A. Initiation of slime mold aggregation viewed as an instability. J Theor Biol, 1970, 26:399-415
[3] Hillen T, Painter K J. A user's guide to PDE models for chemotaxis. J Math Biol, 2009, 58:183-217
[4] Horstmann D. From 1970 until present:the Keller-Segel model in chemotaxis and its consequences. Jahresber DMV, 2003, 105:103-165
[5] Zhong X, Jiang S. Globally bounded in-time solutions to a parabolic-elliptic system modeling chemotaxis. Acta Mathematica Scientia, 2007, 27B:421-429
[6] Ryu S U, Yagi A. Optimal control of Keller-Segel equations. J Math Anal Appl 2001, 256:45-66
[7] Guo B Z, Zhang L. Local null controllability for a chemotaxis system of parabolic-elliptic type. Systems & Control Letters, 2014, 65:106-111
[8] Guo B Z, Zhang L. Local exact controllability to positive trajectory for parabolic system of chemotaxis. Math Control Relat Fields, 2016, 6:143-165
[9] Barbu V. Controllability of parabolic and Navier-Stokes equations. Sci Math Jpn, 2002, 56:143-211
[10] Ammar-Khodja F, Benabdallah A, Gonzalez-Burgos M, etal. Recent results on the controllability of linear coupled parabolic problems:a survey. Math Control Relat Fields, 2011, 1:267-306
[11] Fernández-Cara E, Zuazua E. Null and approximate controllability for weakly blowing up semilinear heat equations. Ann Inst Henri Poincaré, Analyse non linéaire, 2000, 17:583-616
[12] Wang L, Wang G. The Bang-Bang principle of time optimal controls for the heat equation with internal controls. Systems & Control Letters, 2007, 56:709-713
[13] Wang G. L^∞-Null Controllability for the heat equation and its consequences for the time optimal control problem. SIAM J Control & Optimization, 2008, 47:1701-1720
[14] Agmon S, Douglis A, Nirenberg L. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I. Comm Pure Appl Math, 1959, 12:623-727
[15] Ladyzhenskaja O A, Solonnikov V A, Ural'ceva N N. Linear and Quasi-linear Equations of Parabolic Type. Providence:AMS, 1968
[16] DiBenedetto E. Degenerate Parabolic Equations. New York:Springer-Verlag, 1993
[17] Wang L, Wang G. The optimal time control of a phase-field system. SIAM J Control & Optimization, 2006, 42:1483-1508
[18] Barbu V. Analysis and Control of Nonlinear Infinite-Dimensional Systems. Boston:Academic Press, 1993
[19] Rothe F. Global Solutions of Reaction-Diffusion systems. Berlin:Springer-Verlag, 1984