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

Solutions (Formula presented.) to the chemotaxis system (Formula presented.) in a ball (Formula presented.), (Formula presented.), wherein (Formula presented.) and (Formula presented.) are given parameters with m−q>−1, cannot blow up in finite time provided u is uniformly-in-time bounded in (Formula presented.) for some (Formula presented.). For radially symmetric solutions, we show that, if u is only bounded in (Formula presented.) and the technical condition (Formula presented.) is fulfilled, then, for any (Formula presented.), there is C>0 with (Formula presented.) (Formula presented.) denoting the maximal existence time. This is essentially optimal in the sense that, if this estimate held for any (Formula presented.), then u would already be bounded in (Formula presented.) for some (Formula presented.).