Sharp well-posedness and blowup results for parabolic systems of the Keller-Segel type

Piotr Biler (1, 2) , Alexandre Boritchev (3, 4) , Lorenzo Brandolese (5, 4)

1 Institute of Mathematics [Wrocław]
2 UWr - University of Wrocław [Poland]
3 MMCS - Modélisation mathématique, calcul scientifique
4 ICJ - Institut Camille Jordan
5 EDPA - Équations aux dérivées partielles, analyse

We study two toy models obtained after a slight modification of the nonlinearity of the usual doubly parabolic Keller-Segel system. For these toy models, both consisting of a system of two parabolic equations, we establish that for data which are, in a suitable sense, smaller than the diffusion parameter τ in the equation for the chemoattractant, we obtain global solutions, and for some data larger than τ , a finite time blowup. In this way, we check that our size condition for the global existence is sharp for large τ , up to a logarithmic factor.

Domaines: Equations aux dérivées partielles [math.AP]

Piotr Biler, Alexandre Boritchev, Lorenzo Brandolese. Sharp well-posedness and blowup results for parabolic systems of the Keller-Segel type. Methods and Applications of Analysis, 2023, 30 (2), pp.53-76. ⟨10.4310/MAA.2023.v30.n2.a1⟩. ⟨hal-03699868⟩