Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller-Segel system

We consider a parabolic-elliptic chemotaxis system generalizing $$\{\begin{array}{l}\begin{array}{rl}{\displaystyle }& {\displaystyle u_t=\mathrm{\nabla }\cdot ((u+1{)}^{m-1}\mathrm{\nabla }u)-\mathrm{\nabla }\cdot (u(u+1{)}^{\sigma -1}\mathrm{\nabla }v)}\\ {\displaystyle }& {\displaystyle 0=\mathrm{\Delta }v-v+u}\end{array}\end{array}$$ in bounded smooth domains $\Omega\subset $\mathbb{R}$^N$,$N\ge 3$, and with homogeneous Neumann boundary conditions. We show that *) solutions are global and bounded if $\sigma <m-\frac{N-2}{N}$ *) solutions are global if $\sigma \le 0$ *) close to given radially symmetric functions there are many initial data producing unbounded solutions if $\sigma >m-\frac{N-2}{N}$. In particular, if $\sigma \le 0$ and $\sigma >m-\frac{N-2}{N}$, there are many initial data evolving into solutions that blow up after infinite time.