Can diffusion degeneracies enhance complexity in chemotactic aggregation? Finite-time blow-up on spheres in a quasilinear Keller–Segel system

The Cauchy problem in \mathbb{R}^{n} for the cross-diffusion system \begin{cases}u_{t} = \nabla \cdot (D(u)\nabla u) - \nabla\cdot (u\nabla v), \\ 0 = \Delta v +u,\end{cases} is considered for n\ge 2 and under assumptions ensuring that D suitably generalizes the prototype given by D(\xi)=(\xi+1)^{-\alpha}, \quad \xi\ge 0. Under the assumption that \alpha>1 , it is shown that for any r_{\star}>0 and \delta\in (0,1) one can find radially symmetric initial data from C_{0}^{\infty}(\mathbb{R}^{n}) such that the corresponding solution blows up within some finite time, and that this explosion occurs throughout certain spheres in an appropriate sense, with any such sphere being located in the annulus \overline{B}_{r_\star+\delta}(0)\setminus B_{(1-\delta)r_\star}(0) .This is complemented by a result revealing that when \alpha<1 , any finite-mass unbounded radial solution must blow up exclusively at the spatial origin.