Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion

Adrien Blanchet (1) , Philippe Laurencot (2)

1 GREMAQ - Groupe de recherche en économie mathématique et quantitative
2 IMT - Institut de Mathématiques de Toulouse UMR5219

For a specific choice of the diffusion, the parabolic-elliptic Patlak-Keller-Segel system with non-linear diffusion (also referred to as the quasi-linear Smoluchowski-Poisson equation) exhibits an interesting threshold phenomenon: there is a critical mass $M_c>0$ such that all the solutions with initial data of mass smaller or equal to $M_c$ exist globally while the solution blows up in finite time for a large class of initial data with mass greater than $M_c$. Unlike in space dimension $2$, finite mass self-similar blowing-up solutions are shown to exist in space dimension $d≥3$.

Mots clés: Backward self-similar solutions; blowup; chemotaxis; Patlak-Keller-Segel model; degenerate diffusion

Domaines: Equations aux dérivées partielles [math.AP]

Adrien Blanchet, Philippe Laurencot. Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion. Communications on Pure and Applied Mathematics, 2012, 11 (1), pp.47-60. ⟨10.3934/cpaa.2012.11.47⟩. ⟨hal-00429490⟩