Oscillating solutions to a parabolic-elliptic system related to a chemotaxis model

Yūki Naito; Takasi Senba

doi:10.3934/proc.2011.2011.1111

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2011, Volume 2011: 1111-1118. Doi: 10.3934/proc.2011.2011.1111 open access

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Oscillating solutions to a parabolic-elliptic system related to a chemotaxis model

Yūki Naito^1, and
Takasi Senba^2,

1.
Department of Mathematics, Faculty of Sciences, Ehime University, Matsuyama, 790-8577
2.
Faculty of Engineering, Kyushu Institute of Technology, Kitakyushu, 804-8550

Received: June 30, 2010

Revised: December 31, 2010

Published: August 2017

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Abstract

In this paper, we consider solutions to a Cauchy problem for a parabolic-elliptic system in two dimensional space. This system is a simplied version of a chemotaxis model, and is also a model of self-interacting particles. The behavior of solutions to the problem closely depends on the $L^1$-norm of the solutions. If the quantity is larger than 8$\pi$, the solution blows up in nite time. If the quantity is smaller than the critical mass, the solution exists globally in time. In the critical case, innite blowup solutions are found. In the present paper, we direct our attention to radial solutions to the problem whose $L^1$-norm is equal to 8$\pi$ and nd oscillating solutions.

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Mathematics Subject Classification: Primary: 35K55, 35K57; Secondary: 92C17.

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