Large global-in-time solutions  of the parabolic-parabolic  Keller-Segel system on the plane

Piotr Biler; Ignacio Guerra; Grzegorz Karch

doi:10.3934/cpaa.2015.14.2117

Article Contents

2015, Volume 14, Issue 6: 2117-2126. Doi: 10.3934/cpaa.2015.14.2117

This issue Previous Article Next Article Cyclicity of some Liénard Systems

Large global-in-time solutions of the parabolic-parabolic Keller-Segel system on the plane

1.
Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50--384 Wrocław
2.
Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile
3.
Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław

Received: January 31, 2014

Revised: August 31, 2014

Published: October 2015

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Abstract

As it is well known, the parabolic-elliptic Keller-Segel system of chemotaxis on the plane has global-in-time regular nonnegative solutions with total mass below the critical value $8\pi$. Solutions with mass above $8\pi$ blow up in a finite time. We show that the case of the parabolic-parabolic Keller-Segel is different: each mass may lead to a global-in-time-solution, even if the initial data is a finite signed measure. These solutions need not be unique, even if we limit ourselves to nonnegative solutions.

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Mathematics Subject Classification: Primary: 35Q92; Secondary: 35K40.

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References

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