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

Abstract

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 $\Omega \subset \mathbb R^n$, $n \ge 2$, 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 $L^p(\Omega)$ for some $p > p_0 := \frac n2 (1 - (m - q))$. For radially symmetric solutions, we show that, if $u$ is only bounded in $L^{p_0}(\Omega)$ and the technical condition $m > \frac{n-2 p_0}{n}$ 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 $x \in \Omega$ 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 $L^{p}(\Omega)$ 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 $L^{p_0}(\Omega)$.