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Abstract and Applied Analysis
Volume 2006, Article ID 23061, 21 pages

A quasi-linear parabolic system of chemotaxis

1Department of Applied Mathematics, Faculty of Technology, Miyazaki University, 1-1 Gakuen Kibanadai Nishi, Miyazaki-shi 889-2192, Japan
2Department of Mathematical Science, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama-machi, Toyonaka-shi 560-8531, Japan

Received 15 December 2004; Accepted 21 January 2005

Copyright © 2006 Takasi Senba and Takasi Suzuki. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider a quasi-linear parabolic system with respect to unknown functions u and v on a bounded domain of n-dimensional Euclidean space. We assume that the diffusion coefficient of u is a positive smooth function A(u), and that the diffusion coefficient of v is a positive constant. If A(u) is a positive constant, the system is referred to as so-called Keller-Segel system. In the case where the domain is a bounded domain of two-dimensional Euclidean space, it is shown that some solutions to Keller-Segel system blow up in finite time. In three and more dimensional cases, it is shown that solutions to so-called Nagai system blow up in finite time. Nagai system is introduced by Nagai. The diffusion coefficients of Nagai system are positive constants. In this paper, we describe that solutions to the quasi-linear parabolic system exist globally in time, if the positive function A(u) rapidly increases with respect to u.