Asymptotic stability of a decaying solution to the Keller-Segel system of degenerate type

Abstract

We discuss the global behavior of the weak solution of the Keller-Segel system of degenerate type. Asymptotic stability of the Barenblatt-Pattle solution and its convergence rate for the decaying weak solution in $L^1({\mathbb R}^n)$ is shown for the degenerated case $1 <{\alpha} < 2-\frac{2}{n}$. The method is based on the techniques applied to the Fokker-Plank equation due to Carrillo-Toscani [8] deriving from the explicit time decay of the free energy functional and some new estimates for the nonlinear interaction involving the critical type Sobolev inequality. We give the rigorous justification of those procedures via some approximating procedures.