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

We consider a parabolic-elliptic chemotaxis system generalizing \[ \begin{cases}\begin{split} & u_t=\nabla\cdot((u+1)^{m-1}\nabla u)-\nabla \cdot(u(u+1)^{\sigma-1}\nabla v)\\ & 0 = \Delta v - v + u \end{split}\end{cases} \] 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.