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Immediate smoothing and global solutions for initial data in L1 × W1,2 in a Keller–Segel system with logistic terms in 2D

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

Published online by Cambridge University Press:  02 September 2020

Johannes Lankeit Open the ORCID record for Johannes Lankeit [Opens in a new window]
Johannes Lankeit*
Affiliation:
Institut für Mathematik, Universität Paderborn, Warburger Str. 100, 33098Paderborn, Germany ([email protected])
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Abstract

This paper deals with the logistic Keller–Segel model

\[ \begin{cases} u_t = \Delta u - \chi \nabla\cdot(u\nabla v) + \kappa u - \mu u^2, \\ v_t = \Delta v - v + u \end{cases} \]
in bounded two-dimensional domains (with homogeneous Neumann boundary conditions and for parameters χ, κ ∈ ℝ and μ > 0), and shows that any nonnegative initial data (u0, v0) ∈ L1 × W1,2 lead to global solutions that are smooth in $\bar {\Omega }\times (0,\infty )$.

Keywords

ChemotaxisKeller–Segellogisticregularityinitial dataclassical solutions

MSC classification

Primary: 35B65: Smoothness and regularity of solutions
Secondary: 35K45: Initial value problems for second-order parabolic systems 35A09: Classical solutions 35Q92: PDEs in connection with biology and other natural sciences 92C17: Cell movement (chemotaxis, etc.)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 4 , August 2021 , pp. 1204 - 1224
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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