Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model

Abstract

The Keller-Segel system describes the collective motion of cells that are attracted by a chemical substance and are able to emit it. In its simplest form, it is a conservative drift-diffusion equation for the cell density coupled to an elliptic equation for the chemo-attractant concentration. This paper deals with the rate of convergence towards a unique stationary state in self-similar variables, which describes the intermediate asymptotics of the solutions in the original variables. Although it is known that solutions globally exist for any mass less $8\pi$, a smaller mass condition is needed in our approach for proving an exponential rate of convergence in self-similar variables.

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• Decay rates for a class of diffusive-dominated interaction equations
• On the critical mass Patlak-Keller-Segel system for multi-species populations: global existence and infinite time aggregation
• Self-similar solutions of decaying Keller-Segel systems for several populations
• A functional framework for the Keller-Segel system: logarithmic Hardy-Littlewood-Sobolev and related spectral gap inequalities
• On blowup dynamics in the Keller-Segel model of chemotaxis
• Boundedness of large-time solutions to a chemotaxis model with nonlocal and semilinear flux
• Long-time dynamics of classical Patlak-Keller-Segel equation