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

Abstract

Nonnegative solutions of the Neumann initial-boundary value problem for the chemotaxis system \begin{align}\label{prob:star}\tag{$\star$} \begin{cases} u_t = Δu - \nabla \cdot (u \nabla v) + λu - μu^κ, \\\\ 0 = Δv - \overline m(t) + u, \quad \overline m(t) = \frac1{|Ω|} \int_Ωu(\cdot, t) \end{cases} \end{align} in smooth bounded domains $Ω\subset \mathbb R^n$, $n \ge 1$, are known to be global-in-time if $λ\geq 0$, $μ> 0$ and $κ> 2$. In the present work, we show that the exponent $κ= 2$ is actually critical in the four- and higher dimensional setting. More precisely, if \begin{alignat*}{3} \qquad n &\geq 4, &&\quad κ\in (1, 2) \quad &&\text{and} \quad μ> 0 \\\\ \text{or}\qquad n &\geq 5, &&\quad κ= 2 \quad &&\text{and} \quad μ\in \left(0, \frac{n-4}{n}\right), \end{alignat*} for balls $Ω\subset \mathbb R^n$ and parameters $λ\geq 0$, $m_0 > 0$, we construct a nonnegative initial datum $u_0 \in C^0(\overline Ω)$ with $\int_Ωu_0 = m_0$ 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 $κ\in (1, \frac32)$ (and $λ\geq 0$, $μ> 0$). As the corner stone of our analysis, for certain initial data, we prove that the mass accumulation function $w(s, t) = \int_0^{\sqrt[n]{s}} ρ^{n-1} u(ρ, t) \,\mathrm dρ$ 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 $γ$, the function $ϕ(t) = \int_0^{s_0} s^{-γ} (s_0 - s) w(s, t)$ cannot exist globally.