Giunta, V., Hillen, T., Lewis, M.A. et al. (1 more author) (2025) Positivity and global existence for nonlocal advection-diffusion models of interacting populations. Aims Mathematics, 10 (9). pp. 21254-21272. ISSN: 2473-6988
Abstract
We study a broad class of nonlocal advection-diffusion models describing the behaviour of an arbitrary number of interacting species, each moving in response to the nonlocal presence of others. Our model allows for different nonlocal interaction kernels for each species and arbitrarily many spatial dimensions. We prove the global existence of both non-negative weak solutions in any spatial dimension and positive classical solutions in one spatial dimension. These results generalise and unify various existing results regarding existence of nonlocal advection-diffusion equations. We demonstrate that solutions can blow up in finite time when the detection radius becomes zero, i.e. when the system is local, thus showing that nonlocality is essential for the global existence of solutions. We verify our results with numerical simulations on 2D spatial domains.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0) |
Keywords: | nonlocal advection; positivity of PDEs; global existence; blow-up; numerical PDEs |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematical and Physical Sciences |
Funding Information: | Funder Grant number ENGINEERING AND PHYSICAL SCIENCE RESEARCH COUNCIL EP/V002988/1 |
Date Deposited: | 08 Oct 2025 08:13 |
Last Modified: | 08 Oct 2025 08:13 |
Status: | Published |
Publisher: | American Institute of Mathematical Sciences (AIMS) |
Refereed: | Yes |
Identification Number: | 10.3934/math.2025949 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:232656 |