Campus Units

Mathematics

Document Type

Article

Publication Version

Accepted Manuscript

Publication Date

9-2003

Journal or Book Title

Zeitschrift für angewandte Mathematik und Physik

Volume

54

First Page

839

Last Page

868

DOI

10.1007/s00033-003-3206-1

Abstract

In this paper we study finite time blow-up of solutions of a hyperbolic model for chemotaxis. Using appropriate scaling this hyperbolic model leads to a parabolic model as studied by Othmer and Stevens (1997) and Levine and Sleeman (1997). In the latter paper, explicit solutions which blow-up in finite time were constructed. Here, we adapt their method to construct a corresponding blow-up solution of the hyperbolic model. This construction enables us to compare the blow-up times of the corresponding models. We find that the hyperbolic blow-up is always later than the parabolic blow-up. Moreover, we show that solutions of the hyperbolic problem become negative near blow-up. We calculate the “zero-turning-rate” time explicitly and we show that this time can be either larger or smaller than the parabolic blow-up time.

The blow-up models as discussed here and elsewhere are limiting cases of more realistic models for chemotaxis. At the end of the paper we discuss the relevance to biology and exhibit numerical solutions of more realistic models.

Comments

This is a manuscript of an article published as Hillen, T. H. O. M. A. S., and Howard A. Levine. "Blow-up and pattern formation in hyperbolic models for chemotaxis in 1-D." Zeitschrift für Angewandte Mathematik und Physik (ZAMP) 54, no. 5 (2003): 839-868. The final publication is available at Springer via: 10.1007/s00033-003-3206-1. Posted with permission.

Copyright Owner

Birkhauser Verlag, Basel

Language

en

File Format

application/pdf

Published Version

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