Chen, L; Liu, J-G; Wang, J, *Multidimensional Degenerate Keller–Segel System with Critical Diffusion Exponent $2n/(n+2)$*,
Siam Journal on Mathematical Analysis, vol. 44 no. 2
(January, 2012),
pp. 1077-1102 [doi] .
**Abstract:**

*This paper deals with a degenerate diffusion Patlak-Keller-Segel system in n = 3 dimension. The main difference between the current work and many other recent studies on the same model is that we study the diffusion exponent m = 2n/(n + 2), which is smaller than the usual exponent m* = 2-2/n used in other studies. With the exponent m = 2n/(n + 2), the associated free energy is conformal invariant, and there is a family of stationary solutions Uλ,x0 (x) = C(λ/ λ 2+|x-x0| 2 ) n+2/2 λ < 0, σ0 ? ℝn. For radially symmetric solutions, we prove that if the initial data are strictly below Uλ,0(x) for some λ, then the solution vanishes in L1 loc as tλ8; if the initial data are strictly above Uλ,0(x) for some λ, then the solution either blows up at a finite time or has a mass concentration at r = 0 as time goes to infinity. For general initial data, we prove that there is a global weak solution provided that the Lm norm of initial density is less than a universal constant, and the weak solution vanishes as time goes to infinity. We also prove a finite time blow-up of the solution if the Lm norm for initial data is larger than the Lm norm of Uλ,x0 (x), which is constant independent of λ and x0, and the free energy of initial data is smaller than that of Uλ,x0(x). © 2012 Society for Industrial and Applied Mathematics.*