Exponential speed of uniform convergence of the cell density toward equilibrium for subcritical mass in a Patlak–Keller–Segel model

Abstract

This paper is concerned with a chemotaxis aggregation model for cells, more precisely with a parabolic–elliptic semilinear Patlak–Keller–Segel system in a ball of RN, exhibiting a critical mass phenomenon for N≥2. The main result of this paper is the exponential speed of uniform convergence of radial solutions toward the unique steady state in the subcritical case. We stress that this covers in particular the classical Keller–Segel system with N=2, well known for its critical mass 8π, and that the result improves on the known results even for this most studied problem. A key tool is an associated one-dimensional degenerate parabolic problem. The proof exploits its formal gradient flow structure ut=−∇F[u(t)] on an “infinite dimensional Riemannian manifold”. In particular, we show a new Hardy type inequality, equivalent to the strict convexity of F at any steady state of subcritical mass, which heuristically explains the exponential speed of convergence.