Contents Online
Methods and Applications of Analysis
Volume 30 (2023)
Number 2
Sharp well-posedness and blowup results for parabolic systems of the Keller–Segel type
Pages: 53 – 76
DOI: https://dx.doi.org/10.4310/MAA.2023.v30.n2.a1
Authors
Abstract
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 $\tau/(\ln \tau)^3$, where $\tau$ is the diffusion parameter in the equation for the chemo-attractant, we obtain global solutions. Moreover, for a class of data larger than $\tau$, we obtain the finite time blowup, in the whole space as well as in a bounded domain, with two different techniques. Thus, our analysis implies that our size condition on the initial data for the global existence of solutions is sharp, for large $\tau$, up to a logarithmic factor. These results show that global-in-time solutions can be obtained more easily with bigger diffusion coefficient $\tau$, similarly as is known for weaker nonlinear cross-diffusion terms compared to the strength of diffusion in the first equation.
Keywords
chemotaxis, cross-diffusion, Besov spaces, pseudo-measures, global-in-time solutions, blowup
2010 Mathematics Subject Classification
35B40, 35B44, 35K40, 35K55, 35Q92
The first named author would like to thank Institut Camille Jordan, Université Claude Bernard-Lyon 1 for hospitality during his sabbatical stay (Sep 2021–Jan 2022) as a fellow of Institut des Études Avancées – Collegium de Lyon, partially supported by the Polish NCN grant 2016/23/B/ST1/00434.
The authors also gratefully acknowledge the support received by the ARQUS program.
Received 16 September 2022
Accepted 21 June 2023
Published 8 January 2024