The Keller-Segal equation exhibits a competition between diffusion and effects leading to concentration, and depending on whether the total mass is above or below a critical value, one or the other wins. We examine the long time behavior for critical mass making crucial use of a displacement convex Lyapunov functional whose existence is a pleasant surprise -- the equation itself describes gradient flow in the Wasserstein metric, but of a non displacement convex functional. This is joint work with Jose Carrillo and Adrien Blanchet.
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