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Global solvability and explicit bounds for non-local adhesion models

Published online by Cambridge University Press:  23 November 2017

T. HILLEN ,
K. J. PAINTER  and
M. WINKLER
T. HILLEN
Affiliation:
Centre for Mathematical Biology, University of Alberta, Edmonton, Canada email: thillen@ualberta.ca
K. J. PAINTER
Affiliation:
Heriot-Watt University, Edinburgh, UK & Department of Mathematics, Politecnico di Torino, Torino, Italy email: K.Painter@hw.ac.uk
M. WINKLER
Affiliation:
Institut für Mathematik, Universität Paderborn, Paderborn, Germany email: michael.winkler@math.uni-paderborn.de
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Abstract

Adhesion between cells and other cells (cell–cell adhesion) or other tissue components (cell–matrix adhesion) is an intrinsically non-local phenomenon. Consequently, a number of recently developed mathematical models for cell adhesion have taken the form of non-local partial differential equations, where the non-local term arises inside a spatial derivative. The mathematical properties of such a non-local gradient term are not yet well understood. Here we use sophisticated estimation techniques to show local and global existence of classical solutions for such examples of adhesion-type models, and we provide a uniform upper bound for the solutions. Further, we discuss the significance of these results to applications in cell sorting and in cancer invasion and support the theoretical results through numerical simulations.

Keywords

adhesionnon-local PDEexistence and uniquenessuniform boundscancer invasion35Q9235K5792B05

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Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Issue 4 , August 2018 , pp. 645 - 684
Copyright
Copyright © Cambridge University Press 2017 

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