# American Institute of Mathematical Sciences

## Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller-Segel system

Received  August 2017 Revised  October 2017 Published  January 2019

We consider a parabolic-elliptic chemotaxis system generalizing
 \begin{align} & {{u}_{t}}=\nabla \cdot ({{(u+1)}^{m-1}}\nabla u)-\nabla \cdot (u{{(u+1)}^{\sigma -1}}\nabla v) \\ & \ 0=\Delta v-v+u \\ \end{align}
in bounded smooth domains
 $\Omega \subset \mathbb{R}^N$
,
 $N\ge 3$
, and with homogeneous Neumann boundary conditions. We show that
● solutions are global and bounded if
 $\sigma ● solutions are global if $ \sigma\le 0 $● close to given radially symmetric functions there are many initial data producing unbounded solutions if $ \sigma>m-\frac{N-2}{N} $. In particular, if $ \sigma\le 0 $and $ \sigma>m-\frac{N-2}{N} \$
, there are many initial data evolving into solutions that blow up after infinite time.
Citation: Johannes Lankeit. Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller-Segel system. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020013
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