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The parabolic-parabolic Keller-Segel model in \(\mathbb R^2\). (English) Zbl 1149.35360

Summary: 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.

MSC:

35K45 Initial value problems for second-order parabolic systems
35B60 Continuation and prolongation of solutions to PDEs
35Q80 Applications of PDE in areas other than physics (MSC2000)
92C17 Cell movement (chemotaxis, etc.)
92B05 General biology and biomathematics
35B40 Asymptotic behavior of solutions to PDEs
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