2009/02/19 by Tomasz Cieślak, Cieślak, Tomasz, Philippe Laurençot +1
Engineering · Mathematics · Medicine · #35B05 #35B33 #35B45 #35B60 #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Mathematical and Theoretical Epidemiology and Ecology Models #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.0902.3378
openalex publication_date 2009/02/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is known that classical solutions to the one-dimensional quasilinear Smoluchowski-Poisson system with nonlinear diffusion a(u)=(1+u)-p may blow up in finite time if p>1 and exist globally if p<1. The case p=1 thus appears to be critical but it turns out that all solutions are global also in that case. Two classes of diffusion coefficients are actually identified in this paper, one for which all solutions to the corresponding quasilinear Smoluchowski-Poisson system are global and the other one leading to finite time blow-up for sufficiently concentrated initial data. The cornerstone of the proof are an alternative formulation of the Smoluchowski-Poisson system which relies on a novel change of variables and a virial identity.