Boundedness and blow-up for a quasilinear Keller-Segel system with flux limitation and indirect signal production

The quasilinear Keller-Segel system with flux limitation and indirect signal production u_t=\nabla\cdot\left(D(u)\nabla u\right) -\nabla\cdot\left(u(1+\left|\nabla v\right|^2)^σ\nabla v\right), &x\inΩ, t>0, \ 0=Δv-v+w,x\inΩ, t>0, \ w_t=-w+u,x\inΩ, t>0, under homogeneous Neumann boundary conditions in a smooth bounded domain Ω\subsetR^N is considered, where D(u)\simeq u^{m-1} as u\simeq\infty. We conclude that For N=1 and any σ\in$\mathbb{R}$, if m\geq0, the classical solution exists globally, and it is moreover bounded if m>0. However, if m<0 and Ωis a ball, there exist radially symmetric initial data such that the classical solution exhibits finite-time blow-up. For any N\geq2 and m>1-\frac{1}{N}, if σ\leq \frac{mN+2-2N}{2N-2}, the classical solution is global. Furthermore, if σ<\frac{mN+2-2N}{2N-2}, the corresponding solution is uniformly bounded. For any N\geq2 and m<2-\frac{2}{N}, if σ>\max\left{\frac{N}{2-2N},\frac{mN+2-2N}{2N-2}\right} and Ωis a ball, there exist radially symmetric initial data such that the classical solution blows up in finite time.