A family of mass‐critical Keller–Segel systems

The no-flux initial-boundary value problem for the quasilinear Keller–Segel system u t = ∇ · (D (u) ∇ u) − ∇ · (S (u) ∇ v), v t = Δ v − v + u, equation *6pc &array{luₜ= (D (u) u) - (S (u) v), *-6pc\\3pt vₜ= v-v+u, array. } equation (*) is considered in smoothly bounded domains Ω ⊂ R n Rⁿ, n ⩾ 3 n 3, where D ∈ C 2 ([ 0, ∞) ) D C² ([0, ) ) and S ∈ C 2 ([ 0, ∞) ) S C² ([0, ) ) are such that D > 0 D>0 on [ 0, ∞) [0, ) and that S (0) = 0 0 s>0. A particular focus is on cases in which there exist κ > 0, C S D > 0 >0, Cₒ₃>0 and f ∈ L 1 ( (1, ∞) ) f L¹ ( (1, ) ) such that − f (s) ⩽ D (s) S (s) − κ s 2 / n ⩽ C S D s for all s ⩾ 1. equation *6pc- f (s) D (s) S (s) - s^{2/n} Cₒ₃s for all s 1. *-6pc equation (**) It is first shown that then there exists m 0 > 0 m₀>0 such that whenever u 0 u₀ and v 0 v₀ are reasonably regular and nonnegative with ∫ Ω u 0 < m 0 _ u₀ < m₀, within a suitably generalized concept of solvability one can always find a global solution for which u u remains bounded with respect to the norm in L 2 (n − 1) / n (Ω) L^2 (n-1) /{n} (). Second, in radially symmetric settings this is complemented by a converse result on nonexistence of such solutions for some appropriately large initial data, hence leading to the conclusion that any pair (D, S) (D, S) from the family of nonlinearities fulfilling (**) gives rise to a critical mass phenomenon in (*). Remarkably, this does not only include cases of arbitrarily strong diffusion degeneracies due to fast decay of D (s) D (s) as s → ∞ s, but according to the considerably wide funnel described by (*) this moreover indicates that mass criticality in Keller–Segel systems is far more than a nongeneric phenomenon limited to precise functional forms of model ingredients hardly to be found in nature. Applications to concrete scenarios include the detection of mass-critical probability distribution functions of the form 0 ⩽ s ↦ exp (− s (n − 2) / n) 0 s (-s^ (n-2) /{n}) in the version of (**) accounting for so-called volume-filling effects.