Communications in Mathematical Sciences
- Commun. Math. Sci.
- Volume 6, Number 2 (2008), 417-447.
The parabolic-parabolic Keller-Segel model in R2
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.
Commun. Math. Sci., Volume 6, Number 2 (2008), 417-447.
First available in Project Euclid: 1 July 2008
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Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
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