Chemotaxis can prevent thresholds on population density

Johannes Lankeit

doi:10.3934/dcdsb.2015.20.1499

Article Contents

2015, Volume 20, Issue 5: 1499-1527. Doi: 10.3934/dcdsb.2015.20.1499

This issue Previous Article The improved results on the stochastic Kolmogorov system with time-varying delay Next Article Convective nonlocal Cahn-Hilliard equations with reaction terms

Chemotaxis can prevent thresholds on population density

Johannes Lankeit^1,

1.
Institut für Mathematik, Universitat Paderborn, 33098 Paderborn

Received: February 28, 2014

Revised: December 31, 2014

Published: June 2015

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Abstract

We define and (for $q>n$) prove uniqueness and an extensibility property of $W^{1,q}$-solutions to \begin{align*} u_t = -\nabla \cdot (u \nabla v)+\kappa u-\mu u^2\\ 0 = \Delta v - v + u \\ \partial_v v|\partial \Omega = \partial_v u|\partial \Omega = 0 , u(0,\cdot) = u_0, \end{align*} in balls in $\mathbb{R}^n$. They exist globally in time for $\mu\ge 1$ and, for a certain class of initial data, undergo finite-time blow-up if $\mu<1$.
We then use this blow-up result to obtain a criterion guaranteeing some kind of structure formation in a corresponding chemotaxis system - thereby extending recent results of Winkler [26] to the higher dimensional (radially symmetric) case.

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Mathematics Subject Classification: Primary: 35K55; Secondary: 35B44, 35A01, 35A02, 35Q92, 92C17.

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