## Differential and Integral Equations

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

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

#### 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