Differential and Integral Equations

Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems

Yoshie Sugiyama

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Abstract

We consider the degenerate Keller-Segel system (KS) of Nagai type below. We prove that when $m > 2-\frac{2}{N}$, the problem (KS) is solvable globally in time without any restriction on the size of the initial data and that when $1 < m \le 2-\frac{2}{N}$, the problem (KS) evolves in a finite time blow-up for some large initial data. Hence, we completely classify the existence and non-existence of the time global solution by means of the exponent $m=2-\frac{2}{N}$, which generalizes the Fujita exponent for (KS).

Article information

Source
Differential Integral Equations, Volume 19, Number 8 (2006), 841-876.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356050338

Mathematical Reviews number (MathSciNet)
MR2263432

Zentralblatt MATH identifier
1212.35240

Subjects
Primary: 35K45: Initial value problems for second-order parabolic systems
Secondary: 35B40: Asymptotic behavior of solutions 35K57: Reaction-diffusion equations 92C17: Cell movement (chemotaxis, etc.)

Citation

Sugiyama, Yoshie. Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems. Differential Integral Equations 19 (2006), no. 8, 841--876. https://projecteuclid.org/euclid.die/1356050338


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