Regularization in Keller-Segel type systems and the De Giorgi method

Fokker-Planck systems modeling chemotaxis, haptotaxis, and angiogenesis are numerous and have been widely studied. Several results exist that concern the gain of Lp integrability but methods for proving regularizing effects in L∞ are still very few.

Here, we consider a special example, related to the Keller-Segel system, which is both illuminating and singular by lack of diffusion on the second equation (the chemical concentration). We show the gain of L∞ integrability (strong hypercontractivity) when the initial data belongs to the scaleinvariant space.

Our proof is based on De Giorgi’s technique for parabolic equations. We present this technique in a formalism which might be easier that the usual iteration method. It uses an additional continuous parameter and makes the relation to kinetic formulations for hyperbolic conservation laws.

Keywords

De Giorgi method, entropy methods, regularizing effects, hypercontractivity, Keller-Segel system, haptotaxis

2010 Mathematics Subject Classification

35B65, 35K55, 92C17

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Published 21 December 2011