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Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions. (English) Zbl 1172.35035
The paper concerns qualitative properties of non-negative solutions for the Patlak-Keller-Segel system of dimension \(d \geq 3\) with homogeneous nonlinear diffusion. The diffusion is chosen in such a way that its scaling and the one of the Poisson term coincide. It is exhibited that the qualitative behavior of solutions is decided by the initial mass of the system, i.e. there is a sharp critical mass \(M_c\) such that free energy solutions exist globally for \(M \in (0, M_c]\) while there are finite time blowing-up solutions otherwise.
The main results of the work are as follows. If \(0\leq M\leq M_c\) then solutions exist globally in time, and there exists a radially symmetric compactly supported self-similar solution, although the authors could not show that it attracts all global solutions. If \(M=M_c\) then solutions exist globally in time, and there are infinitely many compactly supported stationary solutions. If \(M>M_c\) then it is proved that there exist solutions, corresponding to initial data with negative free energy, blowing up in finite time. However, the possibility that solutions with positive free energy may be global in time cannot be excluded.

MSC:
35K65 Degenerate parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
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