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

Solutions (u,v) to the chemotaxis system ut=∇⋅((u+1)m−1∇u−u(u+1)q−1∇v),τvt=Δv−v+uin a ball Ω⊂Rn, 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 Lp(Ω) for some p>p0:=n2(1−(m−q)). For radially symmetric solutions, we show that, if u is only bounded in Lp0(Ω) and the technical condition m>n−2p0n is fulfilled, then, for any α>np0, there is C>0 with u(x,t)≤C|x|−αfor all x∈Ω and t∈(0,Tmax),Tmax∈(0,∞] denoting the maximal existence time. This is essentially optimal in the sense that, if this estimate held for any αp0.