Large mass self-similar solutions of the parabolic-parabolic Keller--Segel model of chemotaxis

Piotr Biler (1) , Lucilla Corrias (2, 3) , Jean Dolbeault (4)

1 Instytut Matematyczny
2 LAP - Laboratoire Analyse et Probabilités
3 UEVE - Université d'Évry-Val-d'Essonne
4 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

In two space dimensions, the parabolic-parabolic Keller--Segel system shares many properties with the parabolic-elliptic Keller--Segel system. In particular, solutions globally exist in both cases as long as their mass is less than 8π. However, this threshold is not as clear in the parabolic-parabolic case as it is in the parabolic-elliptic case, in which solutions with mass above 8π always blow up. Here we study forward self-similar solutions of the parabolic-parabolic Keller--Segel system and prove that, in some cases, such solutions globally exist even if their total mass is above 8π, which is forbidden in the parabolic-elliptic case.

Mots clés: Keller--Segel model; chemotaxis; self-similar solution; nonlocal parabolic equations; critical mass; existence; blowup

Domaines: Equations aux dérivées partielles [math.AP]

Piotr Biler, Lucilla Corrias, Jean Dolbeault. Large mass self-similar solutions of the parabolic-parabolic Keller--Segel model of chemotaxis. Journal of Mathematical Biology, 2011, 63, pp.1-32. ⟨10.1007/s00285-010-0357-5⟩. ⟨hal-00411913⟩