Communications in Mathematical Sciences
Volume 6 (2008)
The parabolic-parabolic Keller-Segel model in R2
Pages: 417 – 447
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.
chemotaxis; parabolic system; global weak solutions; energy method; Onofri inequality; Hardy-Littlewood-Sobolev inequality
2010 Mathematics Subject Classification
35B60, 92B05, 92C17