On the optimality of upper estimates near blow-up in quasilinear Keller–Segel systems. Issue 9 (13th June 2022) - Fuest, Mario
ABSTRACT: Solutions ( u, v ) to the chemotaxis system 1 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 2 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 .