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Volume 18, Issue 2

# Boundedness of Solutions to a Parabolic-Elliptic Keller–Segel Equation in ℝ2 with Critical Mass

Toshitaka Nagai
• Corresponding author
• Course of General Education (Science), National Institute of Technology, Fukui College, Sabae, Fukui 916-8507, Japan
• Email
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Published Online: 2017-06-21 | DOI: https://doi.org/10.1515/ans-2017-6025

## Abstract

We consider the Cauchy problem for a parabolic-elliptic system in ${ℝ}^{2}$, called the parabolic-elliptic Keller–Segel equation, which appears in various fields in biology and physics. In the critical mass case where the total mass of the initial data is $8\pi$, the unboundedness of nonnegative solutions to the Cauchy problem was shown by Blanchet, Carrillo and Masmoudi [7] under some conditions on the initial data, on the other hand, conditions for boundedness were given by Blanchet, Carlen and Carrillo [6] and López-Gómez, Nagai and Yamada [23]. In this paper, we investigate further the boundedness of nonnegative solutions.

MSC 2010: 35B45; 35K15; 35K55

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Revised: 2017-05-25

Accepted: 2017-05-25

Published Online: 2017-06-21

Published in Print: 2018-04-01

Funding Source: Japan Society for the Promotion of Science

Award identifier / Grant number: JP15KT0019

This work was supported by JSPS KAKENHI, Grant Number JP15KT0019.

Citation Information: Advanced Nonlinear Studies, Volume 18, Issue 2, Pages 337–360, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365,

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