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

Johannes Lankeit

doi:10.3934/dcdss.2020013

Article Contents

2020, Volume 13, Issue 2: 233-255. Doi: 10.3934/dcdss.2020013

This issue Previous Article Boundedness in a quasilinear fully parabolic Keller-Segel system via maximal Sobolev regularity Next Article Decay in chemotaxis systems with a logistic term

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

Johannes Lankeit

Institut für Mathematik, Universität Paderborn, Warburger Str. 100, 33098 Paderborn, Germany

Received: July 31, 2017

Revised: September 30, 2017

Published: February 2020

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Abstract

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<m-\frac{N-2}{N} $

● 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.

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Mathematics Subject Classification: Primary: 92C17, 35Q92, 35A01, 35K55.

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