Finite-time blow-up in a two-dimensional Keller--Segel system with an environmental dependent logistic source

Abstract

The Neumann initial-boundary problem for the chemotaxis system \begin{align} \label{prob:abstract} \tag{$\star$} \begin{cases} u_t = Δu - \nabla \cdot (u \nabla v) + κ(|x|) u - μ(|x|) u^p, \\ 0 = Δv - \frac{m(t)}{|Ω|} + u, \quad m(t) := \int_Ωu(\cdot, t) \end{cases} \end{align} is studied in a ball $\Omega =B_R(0)\subset {\mathbb{R}}^2$, $R>0$ for $p\ge 1$ and sufficiently smooth functions $\kappa ,\mu :[0,R]\to [0,\infty )$. We prove that whenever ${\mu }^{\prime },-{\kappa }^{\prime }\ge 0$ as well as $\mu (s)\le {\mu }_1{s}^{2p-2}$ for all $s\in [0,R]$ and some ${\mu }_1>0$ then for all $m_0>8\pi$ there exists $u_0\in C^0(\overline{\Omega })$ with ${\int }_{\Omega }{u}_0=m_0$ and a solution $(u,v)$ to \eqref{prob:abstract} with initial datum $u_0$ blowing up in finite time. If in addition $\kappa \equiv 0$ then all solutions with initial mass smaller than $8\pi$ are global in time, displaying a certain critical mass phenomenon. On the other hand, if $p>2$, we show that for all $\mu$ satisfying $\mu (s)\ge {\mu }_1{s}^{p-2-\epsilon }$ for all $s\in [0,R]$ and some ${\mu }_1,\epsilon >0$ the system \eqref{prob:abstract} admits a global classical solution for each initial datum $0\le u_0\in C^0(\overline{\Omega })$

16 pages