Refined temporal asymptotics near blow-up points in the planar Keller-Segel system

For the Keller-Segel system $$\left\{\, \begin{aligned} u_t &= \Delta u - \nabla \cdot ( u \nabla v ), \\ v_t &= \Delta v - v + u \end{aligned} \right. \tag{$\star$}$$ posed in a planar domain $\Omega$ with Neumann boundary conditions, the existence of classical solutions blowing up at some finite time $T$ has long been established. In fact, it has been shown that for every blow-up point $x$ the quantity $\int_{B_R(x)\cap\Omega} u(\cdot,t )\ln(u(\cdot, t))$ is unbounded as $t\nearrow T$ for all $R > 0$ even though the global mass of $u$ is always conserved. The present manuscript provides some quantitative information on the behavior of such localized $L\log L$ expressions by asserting the existence of $\delta_0=\delta_0(\Omega)>0$ such that any solution to the Neumann problem for ($\star$) blowing up at time $T\in (0,\infty)$ satisfies $$\limsup_{t\nearrow T} \frac{1}{\ln\frac{T}{T-t}}\int_{B_R(x)\cap\Omega} u(\cdot, t)\ln(u(\cdot, t)) \ge \delta_0 \tag{$\star\star$}$$ for all $R > 0$ at each blow-up point $x$. This confirms a certain universality property of the blow-up mechanism seen in the particular examples of radial collapsing solutions constructed in the seminal work [16], especially also beyond the realm of symmetry; apart from that, along with a consequence of ($\star\star$) on the corresponding asymptotics of similarly localized $L^p$ norms of $u$ for $p\in (1,\infty]$, this provides some extension of a known result on non-degeneracy of blow-up points that has concentrated on the choice $p=\infty$ here.