Blowup and Global Solutions in a Chemotaxis-Growth System

Kyungkeun Kang and Angela Stevens

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Submission date: 24. Feb. 2015
published in: SIAM journal on mathematical analysis, 135 (2016), p. 57-72 
DOI number (of the published article): 10.1016/j.na.2016.01.017
Bibtex
MSC-Numbers: 35B44, 35M33, 35A01, 35Q92, 92Q17
Keywords and phrases: chemotaxis-growth system, blowup, global solutions, hyperbolic-elliptic system, parabolic-elliptic system

Abstract:
We study a Keller-Segel type of system, which includes growth and death of the chemotactic species and an elliptic equation for the chemo-attractant. The problem is considered in bounded domains as well as in the whole space. In case the random motion of the chemotactic species is neglected, a hyperbolic-elliptic problem results, for which we characterize blow-up of solutions in finite time and existence of regular solutions globally in time, in dependence on the systems parameters. For the parabolic-elliptic problem in dimensions three and higher, we establish global existence of regular solutions in a limiting case, which is an extension of the results given by Tello and Winkler in 2007.