Finite-time blow-up in fully parabolic quasilinear Keller–Segel systems with supercritical exponents

authored by

Xinru Cao, Mario Fuest

Abstract

We examine the possibility of finite-time blow-up of solutions to the fully parabolic quasilinear Keller–Segel model (Figure presented.) in a ball Ω⊂Rn with n≥2. Previous results show that unbounded solutions exist for all m,q∈R with m-q<n-2n, which, however, are necessarily global in time if q≤0. It is expected that finite-time blow-up is possible whenever q>0 but in the fully parabolic setting this has so far only been shown when max{m,q}≥1. In the present paper, we substantially extend these findings. Our main results for the two- and three-dimensional settings state that (⋆) admits solutions blowing up in finite time if (Formula presented.) that is, also for certain m, q with max{m,q}<1. As a key new ingredient in our proof, we make use of (singular) pointwise upper estimates for u.

Organisation(s)

Institute of Applied Mathematics

External Organisation(s)

Donghua University

Type

Article

Journal

Calculus of Variations and Partial Differential Equations

Volume

64

ISSN

0944-2669

Publication date

17.02.2025

Publication status

E-pub ahead of print

Peer reviewed

Yes

ASJC Scopus subject areas

Analysis, Applied Mathematics

Electronic version(s)

https://doi.org/10.1007/s00526-025-02944-4 (Access: Open)