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Adhesion and volume filling in one-dimensional population dynamics under Dirichlet boundary condition

Part of: Parabolic equations and systems Equations and systems of special type Representations of solutions

Published online by Cambridge University Press:  08 January 2024

Hyung Jun Choi Open the ORCID record for Hyung Jun Choi [Opens in a new window] ,
Seonghak Kim Open the ORCID record for Seonghak Kim [Opens in a new window]  and
Youngwoo Koh Open the ORCID record for Youngwoo Koh [Opens in a new window]
Hyung Jun Choi
Affiliation:
School of Liberal Arts, Korea University of Technology and Education, Cheonan 31253, Republic of Korea (email: hjchoi@koreatech.ac.kr)
Seonghak Kim
Affiliation:
Department of Mathematics, College of Natural Sciences, Kyungpook National University, Daegu 41566, Republic of Korea (email: shkim17@knu.ac.kr)
Youngwoo Koh
Affiliation:
Department of Mathematics Education, Kongju National University, Kongju 32588, Republic of Korea (email: ywkoh@kongju.ac.kr)
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Abstract

We generalize the one-dimensional population model of Anguige & Schmeiser [1] reflecting the cell-to-cell adhesion and volume filling and classify the resulting equation into the six types. Among these types, we fix one that yields a class of advection-diffusion equations of forward-backward-forward type and prove the existence of infinitely many global-in-time weak solutions to the initial-Dirichlet boundary value problem when the maximum value of an initial population density exceeds a certain threshold. Such solutions are extracted from the method of convex integration by Müller & Šverák [12]; they exhibit fine-scale density mixtures over a finite time interval, then become smooth and identical, and decay exponentially and uniformly to zero as time approaches infinity. TE check: Please check the reference citation in abstract.

Keywords

Population modelforward-backward-forward typeadhesion and volume fillingpartial differential inclusionconvex integration

MSC classification

Primary: 35M13: Initial-boundary value problems for equations of mixed type
Secondary: 35K59: Quasilinear parabolic equations 35D30: Weak solutions 92D25: Population dynamics (general)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 49
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Copyright
Copyright © The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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