Extensibility criterion ruling out gradient blow-up in a quasilinear degenerate chemotaxis system with flux limitation

Abstract

This paper deals with the quasilinear degenerate chemotaxis system with flux limitation \begin{equation*} \begin{cases} u_t = \nabla\cdot\left(\dfrac{u^p \nabla u}{\sqrt{u^2 + |\nabla u|^2}} \right) -\chi \nabla\cdot\left(\dfrac{u^q\nabla v}{\sqrt{1 + |\nabla v|^2}}\right), \\[1mm] 0 = \Delta v - \mu + u \end{cases}\end{equation*} under no-flux boundary conditions in balls $\Omega\subset\mathbb{R}^n$, and the initial condition $u|_{t=0}=u_0$ for a radially symmetric and positive initial data $u_0\in C^3(\overline{\Omega})$, where $\chi>0$ and $\mu:=\frac{1}{|\Omega|}\int_{\Omega}u_0$. Bellomo--Winkler (Comm.\ Partial Differential Equations;2017;42;436--473) proved local existence of unique classical solutions and extensibility criterion ruling out gradient blow-up as well as global existence and boundedness of solutions when $p=q=1$ under some conditions for $\chi$ and $\int_\Omega u_0$. This paper derives local existence and extensibility criterion ruling out gradient blow-up when $p,q\geq 1$, and moreover shows global existence and boundedness of solutions when $p>q+1-\frac{1}{n}$.