Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions

TL;DR

This paper investigates finite-time blowup and global unbounded solutions in higher-dimensional quasilinear Keller-Segel systems, revealing optimal conditions for blowup and unboundedness depending on diffusion and cross-diffusion effects.

Contribution

It establishes optimal finite-time blowup results for nonlinear diffusion systems and identifies conditions for global unbounded solutions in cross-diffusion systems.

Findings

Finite-time blowup occurs under optimal nonlinear diffusion conditions.

Global unbounded solutions exist when chemical sensitivity decays fast enough.

Results highlight differences between nonlinear diffusion and cross-diffusion Keller-Segel systems.

Abstract

In this paper we consider quasilinear Keller-Segel type systems of two kinds in higher dimensions. In the case of a nonlinear diffusion system we prove an optimal (with respect to possible nonlinear diffusions generating explosion in finite time of solutions) finite-time blowup result. In the case of a cross-diffusion system we give results which are optimal provided one assumes some proper non-decay of a nonlinear chemical sensitivity. Moreover, we show that once we do not assume the above mentioned non-decay, our result cannot be as strong as in the case of nonlinear diffusion without nonlinear cross-diffusion terms. To this end we provide an example, interesting by itself, of global-in-time unbounded solutions to the nonlinear cross-diffusion Keller-Segel system with chemical sensitivity decaying fast enough, in a range of parameters in which there is a finite-time blowup result in a corresponding case without nonlinear cross-diffusion.