Solutions to a chemotaxis system with spatially heterogeneous diffusion sensitivity

We consider a parabolic-elliptic Keller-Segel system with spatially dependent diffusion sensitivity \begin{eqnarray*} \left{ \begin{array}{l} u_t = \nabla \cdot (|x|^β\nabla u) - \nabla \cdot (u\nabla v), \[1mm] 0 = Δv - μ+ u, \qquad μ:=\frac{1}{|Ω|} \int\limits_Ωu, \end{array} \right. \qquad \qquad (\star) \end{eqnarray*} under homogeneous Neumann boundary conditions in the ball $\mathrm{\Omega }=B_R(0)\subset {\mathbb{R}}^n$. For $\beta >0$ and radially symmetric Hölder continuous initial data, we prove that there exists a pointwise classical solution to $(\star )$ in $(\mathrm{\Omega }\setminus \{0\})\times (0,T)$ for some $T>0$. For radially decreasing initial data satisfying certain compatibility criteria, this solution is bounded and unique in $(\mathrm{\Omega }\setminus \{0\})\times (0,T^{\ast })$ for some $T^{\ast }>0$. Moreover, for $n\ge 2$ and sufficiently accumulated initial data, there exists no solution $(u,v)$ to $(\star )$ in the sense specified above which is globally bounded in time.