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

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 $\mathrm{\Omega }\subset {\mathbb{R}}^n$, and the initial condition $u{\mathrm{\mid }}_{t=0}=u_0$ 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 ${\int }_{\mathrm{\Omega }}{u}_0$. This paper derives local existence and extensibility criterion ruling out gradient blow-up when $p,q\ge 1$, and moreover shows global existence and boundedness of solutions when $p>q+1-\frac{1}{n}$.