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On a model for phase separation on biological membranes and its relation to the Ohta–Kawasaki equation

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

Published online by Cambridge University Press:  11 March 2019

H. ABELS Open the ORCID record for H. ABELS [Opens in a new window]  and
J. KAMPMANN
H. ABELS*
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany e-mails: helmut.abels@mathematik.uni-regensburg.de; johannes.kampmann@mathematik.uni-regensburg.de
J. KAMPMANN
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany e-mails: helmut.abels@mathematik.uni-regensburg.de; johannes.kampmann@mathematik.uni-regensburg.de
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Abstract

We provide a detailed mathematical analysis of a model for phase separation on biological membranes which was recently proposed by Garcke, Rätz, Röger and the second author. The model is an extended Cahn–Hilliard equation which contains additional terms to account for the active transport processes. We prove results on the existence and regularity of solutions, their long-time behaviour, and on the existence of stationary solutions. Moreover, we investigate two different asymptotic regimes. We study the case of large cytosolic diffusion and investigate the effect of an infinitely large affinity between membrane components. The first case leads to the reduction of coupled bulk-surface equations in the model to a system of surface equations with non-local contributions. Subsequently, we recover a variant of the well-known Ohta–Kawasaki equation as the limit for infinitely large affinity between membrane components.

Keywords

Partial differential equations on surfacesphase separationCahn-Hilliard equationOhta–Kawasaki equationreaction-diffusion systems

MSC classification

Primary: 35K52: Initial-boundary value problems for higher-order parabolic systems
Secondary: 35Q92: PDEs in connection with biology and other natural sciences 92C37: Cell biology
Type
Papers
Information
European Journal of Applied Mathematics , Volume 31 , Issue 2 , April 2020 , pp. 297 - 338
Copyright
© Cambridge University Press 2019 

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