A critical exponent in a quasilinear Keller-Segel system with arbitrarily fast decaying diffusivities accounting for volume-filling effects
In this paper, the authors study a quasilinear Keller-Segel system with arbitrarily fast decaying diffusivities accounting for volume-filling effects endowed with homogeneous Neumann boundary conditions in a smoothly bounded domain \(\mathbb{R}^{n}\), \(n\geq3\) with sufficiently regular functions \(D(u)\) and \(S(u)\). When \(\frac{S(u)}{D(u)}\cong Cu^{\alpha}\) with \(C>0\), they establish some criticality of the value \(\alpha=\frac{2}{n}\) for \(n\geq3\), without any additional assumption on the behavior of \(D(s)\) as \(s\rightarrow\infty\), in particular without requiring any algebraic lower bound for \(D\). Moreover, applied to the Keller-Segel system with volume-filling effect for probability distribution functions of the type \(Q(s)=\exp(-s^{\beta})\), \(s\geq0\), for global solvability the exponent \(\beta=\frac{n-2}{n}\) is seen to be critical.\N\NThese results are undoubtedly novel and intriguing, and they extend previous understanding in a seamless manner. 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.
scientific article; zbMATH DE number 7879408
scientific article