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Advanced Nonlinear Studies

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


We consider the Cauchy problem for a parabolic-elliptic system in 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 8π, 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.

Keywords: Parabolic-Elliptic System; Keller–Segel Equation; Chemotaxis System; Critical Mass, Boundedness of Solutions

MSC 2010: 35B45; 35K15; 35K55


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About the article

Received: 2017-05-09

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, DOI: https://doi.org/10.1515/ans-2017-6025.

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