# Multidimensional degenerate Keller-Segel system with critical diffusion exponent 2n/(n + 2)

Published

Journal Article

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.

### Full Text

### Duke Authors

### Cited Authors

- Chen, L; Liu, JG; Wang, J

### Published Date

- May 28, 2012

### Published In

### Volume / Issue

- 44 / 2

### Start / End Page

- 1077 - 1102

### International Standard Serial Number (ISSN)

- 0036-1410

### Digital Object Identifier (DOI)

- 10.1137/110839102

### Citation Source

- Scopus