Communications in Mathematical Sciences

The parabolic-parabolic Keller-Segel model in R2

V. Calvez and L. Corrias

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Abstract

This paper is devoted mainly to the global existence problem for the two-dimensional parabolic-parabolic Keller-Segel system in the full space. We derive a critical mass threshold below which global existence is ensured. Carefully using energy methods and ad hoc functional inequalities, we improve and extend previous results in this direction. The given threshold is thought to be the optimal criterion, but this question is still open. This global existence result is accompanied by a detailed discussion on the duality between the Onofri and the logarithmic Hardy-Littlewood-Sobolev inequalities that underlie the following approach.

Article information

Source
Commun. Math. Sci., Volume 6, Number 2 (2008), 417-447.

Dates
First available in Project Euclid: 1 July 2008

Permanent link to this document
https://projecteuclid.org/euclid.cms/1214949930

Mathematical Reviews number (MathSciNet)
MR2433703

Zentralblatt MATH identifier
1149.35360

Subjects
Primary: 35B60: Continuation and prolongation of solutions [See also 58A15, 58A17, 58Hxx] 35Q80: PDEs in connection with classical thermodynamics and heat transfer 92C17: Cell movement (chemotaxis, etc.) 92B05: General biology and biomathematics

Keywords
chemotaxis parabolic system global weak solutions energy method Onofri inequality Hardy-Littlewood-Sobolev inequality

Citation

Calvez, V.; Corrias, L. The parabolic-parabolic Keller-Segel model in R2. Commun. Math. Sci. 6 (2008), no. 2, 417--447. https://projecteuclid.org/euclid.cms/1214949930


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