On the optimality of upper estimates near blow-up in quasilinear Keller–Segel systems.

Solutions (u , v) to the chemotaxis system u t = ∇ ⋅ ((u + 1) m − 1 ∇ u − u (u + 1) q − 1 ∇ v) , τ v t = Δ v − v + u in a ball Ω ⊂ R n , n ≥ 2 , wherein m , q ∈ R and τ ∈ { 0 , 1 } are given parameters with m−q>−1, cannot blow up in finite time provided u is uniformly-in-time bounded in L p (Ω) for some p > p 0 := n 2 (1 − (m − q)). For radially symmetric solutions, we show that, if u is only bounded in L p 0 (Ω) and the technical condition m > n − 2 p 0 n is fulfilled, then, for any α > n p 0 , there is C>0 with u (x , t) ≤ C | x | − α f o r a l l x ∈ Ω a n d t ∈ (0 , T max) , T max ∈ (0 , ∞ ] denoting the maximal existence time. This is essentially optimal in the sense that, if this estimate held for any α < n p 0 , then u would already be bounded in L p (Ω) for some p > p 0 . [ABSTRACT FROM AUTHOR]