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 $\Omega \subset {\mathbb{R}}^n$, $n\ge 1$, are known to be global-in-time if $\lambda \ge 0$, $\mu >0$ 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 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 $\Omega \subset {\mathbb{R}}^n$ and parameters $\lambda \ge 0$, $m_0>0$, 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 $\kappa \in (1,\frac{3}{2})$ (and $\lambda \ge 0$, $\mu >0$). 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 $\varphi (t)={\int }_0^{s_0}{s}^{-\gamma }(s_0-s)w(s,t)$ cannot exist globally.

13 pages

References in corpus (1)

• Relaxed parameter conditions for chemotactic collapse in logistic-type parabolic-elliptic Keller-Segel systems

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