Solutions to a chemotaxis system with spatially heterogeneous diffusion sensitivity

We consider a parabolic-elliptic Keller-Segel system with spatially dependent diffusion sensitivity {eqnarray*} \{ {array}{l} u_t = ∇ · (|x|^β ∇ u) - ∇ · (u∇ v), \\[1mm] 0 = Δ v - μ + u, μ:=1/|Ω| ∫_Ω u, {array} . () {eqnarray*} under homogeneous Neumann boundary conditions in the ball Ω=BR(0)⊂ Rⁿ. For β>0 and radially symmetric H\"older continuous initial data, we prove that there exists a pointwise classical solution to () in (Ω \0\)× (0,T) for some $T>0$. For radially decreasing initial data satisfying certain compatibility criteria, this solution is bounded and unique in (Ω \0\)× (0,T^*) for some T^*>0. Moreover, for n ≥ 2 and sufficiently accumulated initial data, there exists no solution $(u,v)$ to () in the sense specified above which is globally bounded in time.