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

Solutions $(u,v)$ to the chemotaxis system \begin{align*} \begin{cases} u_t = \nabla \cdot ( (u+1)^{m-1} \nabla u - u (u+1)^{q-1} \nabla v), \ \tau v_t = \Delta v - v + u \end{cases} \end{align*} in a ball $\mathrm{\Omega }\subset {\mathbb{R}}^n$, $n\ge 2$, wherein $m,q\in \mathbb{R}$ and $\tau \in \{0,1\}$ are given parameters with $m-q>-1$, cannot blow up in finite time provided $u$ is uniformly-in-time bounded in $L^p(\mathrm{\Omega })$ for some $p>p_0:=\frac{n}{2}(1-(m-q))$. For radially symmetric solutions, we show that, if $u$ is only bounded in $L^{p_0}(\mathrm{\Omega })$ and the technical condition $m>\frac{n-2p_0}{n}$ is fulfilled, then, for any $\alpha >\frac{n}{p_0}$, there is $C>0$ with \begin{align*} u(x, t) \leq C |x|^{-\alpha} \qquad \text{for all $x\in \mathrm{\Omega }$ and $t \in (0, T_{\max})}, \end{align*}T_{\max} \in (0, \infty]$ denoting the maximal existence time. This is essentially optimal in the sense that, if this estimate held for any $\alpha <\frac{n}{p_0}$, then $u$ would already be bounded in $L^p(\mathrm{\Omega })$ for some $p>p_0$. Moreover, we also give certain upper estimates for chemotaxis systems with nonlinear signal production, even without any additional boundedness assumptions on $u$. The proof is mainly based on deriving pointwise gradient estimates for solutions of the Poisson or heat equation with a source term uniformly-in-time bounded in $L^{p_0}(\mathrm{\Omega })$.