Finite-time blow-up in a quasilinear system of chemotaxis

2008/04/21 by Tomasz Cieślak, Michael Winkler · 7 citations
Biochemistry, Genetics and Molecular Biology · Mathematics · #Cancer Genomics and Diagnostics #Gene Regulatory Network Analysis #Mathematical Biology Tumor Growth

paper · doi:10.1088/0951-7715/21/5/009

openalex publication_date 2008/04/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We consider an elliptic–parabolic system of the Keller–Segel type which involves nonlinear diffusion. We find a critical exponent of the nonlinearity in the diffusion, measuring the strength of diffusion at points of high (population) densities, which distinguishes between finite-time blow-up and global-in-time existence of uniformly bounded solutions. This critical exponent depends on the space dimension n ⩾ 1, where apart from the physically relevant cases n = 2 and n = 3 also the result obtained in the one-dimensional setting might be of mathematical interest: here, namely, finite-time explosion of solutions occurs although the Lyapunov functional associated with the system is bounded from below. Additionally this one-dimensional case is an example to show that L ∞ estimates of solutions to non-uniformly parabolic drift–diffusion equations cannot be expected even when boundedness of the gradient of the drift term is presupposed.

Citations

• Partial Differential Equations
• On explosions of solutions to a system of partial differential equations modelling chemotaxis

Cited by

• A new result for the global existence (and boundedness), regularity and stabilization of a three-dimensional Keller-Segel-Navier-Stokes system modeling coral fertilization
• Boundedness and finite-time blow-up in a quasilinear parabolic-elliptic chemotaxis system with logistic source and nonlinear production
• The fast signal diffusion limit in Keller-Segel(-fluid) systems
• Critical mass for finite-time chemotactic collapse in the critical dimension via comparison
• Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system
• Boundedness in a quasilinear fully parabolic Keller-Segel system of higher dimension with logistic source
• Boundedness and large time behavior in a higher-dimensional Keller--Segel system with singular sensitivity and logistic source