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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

Published online by Cambridge University Press:  02 September 2020

Johannes Lankeit*
Affiliation:
Institut für Mathematik, Universität Paderborn, Warburger Str. 100, 33098Paderborn, Germany (jlankeit@math.uni-paderborn.de)

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 $\bar {\Omega }\times (0,\infty )$.

Type
Research Article
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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