Global existence and boundedness in a chemorepulsion system with superlinear diffusion

Marcel Freitag

doi:10.3934/dcds.2018258

Article Contents

2018, Volume 38, Issue 11: 5943-5961. Doi: 10.3934/dcds.2018258

This issue Previous Article The diffusion phenomenon for damped wave equations with space-time dependent coefficients Next Article

Global existence and boundedness in a chemorepulsion system with superlinear diffusion

Marcel Freitag^,

Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany

* Corresponding author: Marcel Freitag
* Corresponding author: Marcel Freitag

Received: April 30, 2018

Revised: May 31, 2018

Published: August 2018

The author is supported by the Deutsche Forschungsgemeinschaft

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Abstract

In a bounded domain $\Omega\subset\mathbb{R}^n$, where $n\ge 3$, we consider the quasilinear parabolic-parabolic Keller-Segel system

$\begin{equation*}\begin{cases}u_t = \nabla\cdot({D(u)\nabla u+u\nabla v}) \;\;\; &\text{in}\ \Omega\times(0,\infty)\\v_t = \Delta v-v+u &\text{in}\ \Omega\times(0,\infty)\end{cases}\end{equation*}$

with homogeneous Neumann boundary conditions. We will find that the condition $D(u)\geq Cu^{m-1}$ suffices to prove the uniqueness and global existence of solutions along with their boundedness if $D(0)>0$ and $m>1+\frac{(n-2)(n-1)}{n^2}$ which is a very different result from what we know about the same system with nonnegative sensitivity. In the case of degenerate diffusion ($D(0) = 0$) and for the same parameters, locally bounded global weak solutions will be established.

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Mathematics Subject Classification: 35A09, 35B44, 35K35, 35B33, 92C17.

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