Behavior in time of solutions of a Keller-Segel system with flux limitation and source term (Q6095309)

In this paper, the authors studied radially symmetric solutions of a parabolic-elliptic cross-diffusion system with flux limitation and source term. Under smallness conditions on the parameters, the authors proved that the solution \(u(x,t)\) blows up in \(L^{\infty}\)-norm at finite time \(T_{\max}\) and for some \(p>1\) it blows up also in \(L^p\)-norm. In addition a lower bound of blow-up time is derived. Finally, under largeness conditions on the parameters, they showed that the solution is global and bounded in time. These results are undoubtedly novel and intriguing. The derivation is presented and organized exceptionally well, building upon existing ideas from the literature and appropriately citing the relevant references. Moreover, the work introduces significant and nontrivial new ideas. Overall, in my opinion, this is an exemplary piece of research.