Finite time blowup for the parabolic–parabolic Keller–Segel system with critical diffusion

  • Philippe Laurençot

    Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, F-31062 Toulouse Cedex 9, France
  • Noriko Mizoguchi

    Department of Mathematics, Tokyo Gakugei University, Koganei, Tokyo 184-8501, Japan

Abstract

The present paper is concerned with the parabolic–parabolic Keller–Segel system

with degenerate critical diffusion in space dimension , the underlying domain Ω being either or the open ball of with suitable boundary conditions. It has remained open whether there exist solutions blowing up in finite time, the existence of such solutions being known for the parabolic–elliptic reduction with the second equation replaced by . Assuming that and , we prove that radially symmetric solutions with negative initial energy blow up in finite time in and in under mixed Neumann–Dirichlet boundary conditions. Moreover, if and Neumann boundary conditions are imposed on both u and v, we show the existence of a positive constant C depending only on N, Ω, and the mass of such that radially symmetric solutions blow up in finite time if the initial energy does not exceed −C. The criterion for finite time blowup is satisfied by a large class of initial data.

Cite this article

Philippe Laurençot, Noriko Mizoguchi, Finite time blowup for the parabolic–parabolic Keller–Segel system with critical diffusion. Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), no. 1, pp. 197–220

DOI 10.1016/J.ANIHPC.2015.11.002