A critical blow-up exponent for flux limiation in a Keller-Segel system

The parabolic-elliptic cross-diffusion system \[ \left\{ \begin{array}{l} u_t = \Delta u - \nabla \cdot \Big(uf(|\nabla v|^2) \nabla v \Big), \\[1mm] 0 = \Delta v - \mu + u, \qquad \int_\Omega v=0, \qquad \mu:=\frac{1}{|\Omega|} \int_\Omega u dx, \end{array} \right. \] is considered along with homogeneous Neumann-type boundary conditions in a smoothly bounded domain $\Omega\subset R^n$, $n\ge 1$, where $f$ generalizes the prototype given by \[ f(\xi) = (1+\xi)^{-\alpha}, \qquad \xi\ge 0, \qquad \mbox{for all } \xi\ge 0, \] with $\alpha\in R$. In this framework, the main results assert that if $n\ge 2$, $\Omega$ is a ball and \[ \alpha<\frac{n-2}{2(n-1)}, \] then throughout a considerably large set of radially symmetric initial data, an associated initial value problem admits solutions blowing up in finite time with respect to the $L^\infty$ norm of their first components. This is complemented by a second statement which ensures that in general and not necessarily symmetric settings, if either $n=1$ and $\alpha\in R$ is arbitrary, or $n\ge 2$ and $\alpha>\frac{n-2}{2(n-1)}$, then any explosion is ruled out in the sense that for arbitrary nonnegative and continuous initial data, a global bounded classical solution exists.