Symmetrization techniques on unbounded domains: Application to a chemotaxis system on R-N

Abstract

The authors study the parabolic-elliptic system on RN: ∂u/∂t=∇⋅(∇u−χu∇v), 0=Δv−γv+αu, u(0,⋅)=u0, a version of the mathematical model of chemotaxis proposed by Keller and Segel.
A differential inequality for the quantity ∫s0u∗(t,σ)dσ, where u∗ is the decreasing rearrangement of the solution u(t,⋅) with respect to the spatial variable, is obtained.
As a consequence, they obtain e.g. Lp-bounds of the solution (u,v) on R2 and global-in-time existence of solutions under the condition αχ∫R2u0<8π. This result is sharp. It is also proved that if u0 is radially symmetric and αχ∫R2u0>8π, then the solution (u,v) blows up in a finite time. Compared to the previous work of Díaz Díaz and Nagai [Adv. Math. Sci. Appl. 5 (1995), no. 2, 659--680; MR1361010 (96j:35246)], where this problem has been considered on bounded domains of RN, there are some additional technical difficulties connected with the regularity of the derivative ∂u∗/∂t.