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Does indirectness of signal production reduce the explosion-supporting potential in chemotaxis–haptotaxis systems? Global classical solvability in a class of models for cancer invasion (and more)

Part of: Equations of mathematical physics and other areas of application Parabolic equations and systems Physiological, cellular and medical topics Partial differential equations

Published online by Cambridge University Press:  17 July 2020

CHRISTINA SURULESCU Open the ORCID record for CHRISTINA SURULESCU [Opens in a new window]  and
MICHAEL WINKLER
CHRISTINA SURULESCU
Affiliation:
Technische Universität Kaiserslautern, Felix-Klein-Zentrum für Mathematik, 67663 Kaiserslautern, Germany email: surulescu@mathematik.uni-kl.de
MICHAEL WINKLER
Affiliation:
Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany email: michael.winkler@math.uni-paderborn.de
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Abstract

We propose and study a class of parabolic-ordinary differential equation models involving chemotaxis and haptotaxis of a species following signals indirectly produced by another, non-motile one. The setting is motivated by cancer invasion mediated by interactions with the tumour microenvironment, but has much wider applicability, being able to comprise descriptions of biologically quite different problems. As a main mathematical feature constituting a core difference to both classical Keller–Segel chemotaxis systems and Chaplain–Lolas type chemotaxis–haptotaxis systems, the considered model accounts for certain types of indirect signal production mechanisms. The main results assert unique global classical solvability under suitably mild assumptions on the system parameter functions in associated spatially two-dimensional initial-boundary value problems. In particular, this rigorously confirms that at least in two-dimensional settings, the considered indirectness in signal production induces a significant blow-up suppressing tendency also in taxis systems substantially more general than some particular examples for which corresponding effects have recently been observed.

Keywords

chemotaxishaptotaxisindirect signal productionglobal existence

MSC classification

Primary: 35Q92: PDEs in connection with biology and other natural sciences
Secondary: 35B44: Blow-up 35K55: Nonlinear parabolic equations 92C17: Cell movement (chemotaxis, etc.) 35A01: Existence problems
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Issue 4 , August 2021 , pp. 618 - 651
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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