Multi-dimensional Degenerate Keller-Segel System With A Diffusion Exponent (n+2)q-4/n+2

This paper deals with a degenerate diffusion Patlak-Keller-Segel system in n≥ dimension where m=((n+2)q-4)/(n+2)>1,q>2.With the exponent m=((n+2)q-4)/(n+2), there is a family of positive stationary solution Uλ,x0(|x|)= 0, x0∈Rn,and the associated free energy is conformal invariant.For radially symmetric solutions,we prove that if the initial data is strictly below Uλ,x0(|x|) for some λ,then the solution vanishes in Lloc1 as tâ†'∞;if the initial data is strictly above Uλ,x0(|x|) for some λ,then the solution blows up at a finite time or has a mass concentrates 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-q+2 norm of initial density is less than a universal constant,and the weak solution vanishes as time goes to infinity.