Abstract and Applied Analysis

A quasi-linear parabolic system of chemotaxis

Takasi Senba and Takasi Suzuki

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Abstract

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 .

Article information

Source
Abstr. Appl. Anal., Volume 2006 (2006), Article ID 23061, 21 pages.

Dates
First available in Project Euclid: 30 January 2007

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1170202966

Digital Object Identifier
doi:10.1155/AAA/2006/23061

Mathematical Reviews number (MathSciNet)
MR2211660

Zentralblatt MATH identifier
1134.35059

Citation

Senba, Takasi; Suzuki, Takasi. A quasi-linear parabolic system of chemotaxis. Abstr. Appl. Anal. 2006 (2006), Article ID 23061, 21 pages. doi:10.1155/AAA/2006/23061. https://projecteuclid.org/euclid.aaa/1170202966


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