Asymptotic decay for the solutions of the parabolic-parabolic Keller-Segel chemotaxis system in critical spaces

  • Authors:

  • Lucilla Corrias;Benot Perthame

  • Affiliations:

  • Département de Mathématiques, Université d'Evry Val d'Essonne, Rue du Père Jarlan, F 91025 Evry Cedex, France;Département de Mathématiques et Applications, ícole Normale Supérieure, CNRS UMR8553, 45 rue d'Ulm, F 75230 Paris cedex 05, France

  • Venue:
  • Mathematical and Computer Modelling: An International Journal
  • Year:
  • 2008

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Abstract

We consider the classical parabolic-parabolic Keller-Segel system describing chemotaxis, i.e., when both the evolution of the biological population and the chemoattractant concentration are described by a parabolic equation. We prove that when the equation is set in the whole space R^d and dimension d=3 the critical spaces for the initial bacteria density and the chemical gradient are respectively L^a(R^d), ad/2, and L^d(R^d). For in these spaces, we prove that small initial data give rise to global solutions that vanish as the heat equation for large times and that exhibit a regularizing effect of hypercontractivity type.