A blow-up result for a quasilinear chemotaxis system with logistic source in higher dimensions
A quasilinear parabolic-elliptic system of chemotaxis \(u_t=\nabla\cdot(D(u)\nabla u-\chi u\nabla v) +f(u)\), \(\Delta v -\frac{1}{|\Omega|}\int_\Omega u(x,t)dx+u=0\) is studied with a power-like diffusion coefficient \(D(u)\geq D_0u^m\), and a logistic type source \(f(u)=au-bu^\kappa\) in a bounded domain \(\Omega\subset \mathbb R^n\). The authors prove a global in time existence result under some assumptions on \(m\), \(\kappa\), \(n\). Moreover, when \(n\geq 5\), \(m\in[0,\frac{4}{n}-1)\), and \(\kappa\in\left(1,\frac{(3+m)n-2}{2n-2}\right)\), there exist radially symmetric initial data in the ball of arbitrarily prescribed mass such that the corresponding solutions blow up in a finite time. This is an extension of result of \textit{M. Winkler} in [J. Math. Anal. Appl. 384, No. 2, 261--272 (2011; Zbl 1241.35028)] to the quasilinear case, and gives a correction to the arguments in [\textit{P. Zheng} et al., Discrete Contin. Dyn. Syst. 35, No. 5, 2299--2323 (2015; Zbl 1307.35067)].
scientific article; zbMATH DE number 6867356
scientific article