Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems

We consider the no-flux initial-boundary value problem for Keller-Segel-type chemotaxis growth systems of the form

$\begin{eqnarray*} ≤\left\{ \begin{array}{ll} u_t=Δ u -χ \nabla · (u\nabla v) + ρ u -μ u^2, & x∈Ω, \ t>0, \\ v_t=Δ v -v+u, & x∈Ω, \ t>0, \end{array} \right. \end{eqnarray*}$

in a ball $Ω\subset\mathbb{R}^n$ , $n≥ 3$ , with parameters $χ>0, ρ≥ 0$ and $μ>0$ .

By means of an argument based on a conditional quasi-energy inequality, it is firstly shown that if $χ=1$ is fixed, then for any given $K>0$ and $T>0$ one can find radially symmetric initial data, possibly depending on $K$ and $T$ , such that for arbitrary $μ∈ (0, 1)$ the corresponding local-in-time classical solution $(u, v)$ satisfies

$\begin{eqnarray*} u(x, t) > \frac{K}{μ} \end{eqnarray*}$

with some $x∈Ω$ and $t∈ (0, T)$ ; in fact, this growth phenomenon is actually identified as being generic in the sense that the set of all initial data having this property is dense in the set of all suitably regular radial initial data in a certain topology.

Secondly, turning a focus on possible effects of large chemotactic sensitivities, on the basis of the above it is shown that when $ρ≥ 0$ and $μ>0$ are fixed, then for all $L>0, T>0$ and $χ>μ$ one can fix radial initial data $(u_{0, χ}, v_{0, χ})$ which decay in $L^∞(Ω)× W^{1, ∞}(Ω)$ as $χ\to∞$ , and which are such that for the respective solution $(u_χ, v_χ)$ there exist $x∈Ω$ and $t∈ (0, T)$ fulfilling

$\begin{eqnarray*} u_χ(x, t) > L. \end{eqnarray*}$