Approaching optimality in blow-up results for Keller-Segel systems with logistic-type dampening

Nonnegative solutions of the Neumann initial-boundary value problem for the chemotaxis system \begin{align}\label{prob:star}\tag{$\star$} \begin{cases} u_t = \Delta u - \nabla \cdot (u \nabla v) + \lambda u - \mu u^\kappa, \\\\ 0 = \Delta v - \overline m(t) + u, \quad \overline m(t) = \frac1{|\Omega|} \int_\Omega u(\cdot, t) \end{cases} \end{align} in smooth bounded domains $\Omega \subset \mathbb R^n$, $n \ge 1$, are known to be global-in-time if $\lambda \geq 0$, $\mu > 0$ and $\kappa > 2$. In the present work, we show that the exponent $\kappa = 2$ is actually critical in the four- and higher dimensional setting. More precisely, if \begin{alignat*}{3} \qquad n &\geq 4, &&\quad \kappa \in (1, 2) \quad &&\text{and} \quad \mu > 0 \\\\ \text{or}\qquad n &\geq 5, &&\quad \kappa = 2 \quad &&\text{and} \quad \mu \in \left(0, \frac{n-4}{n}\right), \end{alignat*} for balls $\Omega \subset \mathbb R^n$ and parameters $\lambda \geq 0$, $m_0 > 0$, we construct a nonnegative initial datum $u_0 \in C^0(\overline \Omega)$ with $\int_\Omega u_0 = m_0$ for which the corresponding solution $(u, v)$ of \eqref{prob:star} blows up in finite time. Moreover, in 3D, we obtain finite-time blow-up for $\kappa \in (1, \frac32)$ (and $\lambda \geq 0$, $\mu > 0$). As the corner stone of our analysis, for certain initial data, we prove that the mass accumulation function $w(s, t) = \int_0^{\sqrt[n]{s}} \rho^{n-1} u(\rho, t) \,\mathrm d\rho$ fulfills the estimate $w_s \le \frac{w}{s}$. Using this information, we then obtain finite-time blow-up of $u$ by showing that for suitably chosen initial data, $s_0$ and $\gamma$, the function $\phi(t) = \int_0^{s_0} s^{-\gamma} (s_0 - s) w(s, t)$ cannot exist globally.