MULTIDIMENSIONAL DEGENERATE KELLER-SEGEL SYSTEM WITH CRITICAL DIFFUSION EXPONENT 2n/(n+2)

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 U-lambda,U-x0(x) = C(lambda/lambda(2)vertical bar vertical bar x x(0)vertical bar(2))(n+2/2) for all lambda > 0, x(0) is an element of R-n. For radially symmetric solutions, we prove that if the initial data are strictly below U-lambda,U-0(x) for some lambda, then the solution vanishes in L-loc(1) as t -> infinity; if the initial data are strictly above U-lambda,U-0(x) for some lambda, 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 L-m 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 L-m norm for initial data is larger than the L-m norm of U-lambda,U-x0 (x), which is constant independent of lambda and x(0), and the free energy of initial data is smaller than that of U-lambda,U-x0 (x).