Communications on Pure and Applied Analysis (CPAA)

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

Pages: 2117 - 2126, Volume 14, Issue 6, November 2015      doi:10.3934/cpaa.2015.14.2117

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Piotr Biler - Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50--384 Wrocław, Poland (email)
Ignacio Guerra - Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Chile (email)
Grzegorz Karch - Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland (email)

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.

Keywords:  Chemotaxis, parabolic-parabolic Keller-Segel model, large global-in-time solutions.
Mathematics Subject Classification:  Primary: 35Q92; Secondary: 35K40.

Received: February 2014;      Revised: September 2014;      Available Online: September 2015.