Exact criterion for global existence and blow up to a degenerate Keller-Segel system

Li Chen ; Jinhuan Wang

Documenta mathematica, Tome 19 (2014), pp. 103-120

Résumé

A degenerate Keller-Segel system with diffusion exponent m with n+22n​2−n2​ in multi dimension is studied. An exact criterion for global existence and blow up of solution is obtained. The estimates on Ln+22n​ norm of the solution play important roles in our analysis. These estimates are closely related to the optimal constant in the Hardy- Littlewood- Sobolev inequality. In the case of initial free energy less than a universal constant which depends on the inverse of total mass, there exists a constant such that if the Ln+22n​ norm of initial data is less than this constant, then the weak solution exists globally; if the Ln+22n​ norm of initial data is larger than the same constant, then the solution must blow-up in finite time. Our result shows that the total mass, which plays the deterministic role in two dimension case, might not be an appropriate criterion for existence and blow up discussion in multi-dimension, while the Ln+22n​ norm of the initial data and the relation between initial free energy and initial mass are more important.

Détail

DOI : 10.4171/dm/441

Classification : 34A34, 35A01
Mots-clés : blow-up, global existence, nonlinear diffusion, nonlocal aggregation