Concentration phenomena to a chemotaxis system with indirect signal production

TL;DR

This paper investigates a chemotaxis system with no finite-time blow-up, demonstrating infinite-time concentration phenomena at the origin, with solutions approaching a delta function, and explores conditions influencing the mass of this singularity.

Contribution

It establishes the occurrence of infinite-time concentration in a chemotaxis model with bounded solutions, and links the singularity mass to Lyapunov functional properties, contrasting with Keller--Segel system behavior.

Findings

Solutions concentrate at the origin as a delta function in infinite time.

The singularity mass exceeds 8π under certain Lyapunov conditions.

Uniform boundedness of an energy functional is key to the analysis.

Abstract

We consider a parabolic-ODE-parabolic chemotaxis system with radially symmetric initial data in a two-dimensional disk under the $0$-Neumann boundary condition. Although our system shares similar mathematical structures as the Keller--Segel system, the remarkable characteristic of the system we consider is that its solutions cannot blow up in finite time. In this paper, focusing on blow-up solutions in infinite time, we confirm concentration phenomena at the origin. It is shown that the radially symmetric solutions of our system have a singularity like a Dirac delta function in infinite time. This means that there exist a time sequence $\{t_k\}$, a weight $m \ge 8\pi$, and a nonnegative function $f \in L^1(\Omega)$ such that \begin{align*} u(\cdot,t_k) \stackrel{*}{\rightharpoonup} m \delta (0) + f\ \mathrm{as}\ t_k \to \infty. \end{align*} We highlight this result is obtained by showing uniform-in-time boundedness of some energy functional. Moreover, we study whether $m = 8\pi$ or $m > 8\pi$, which is an open problem in the Keller--Segel system. It is proved that the weight $m$ of a delta function singularity is larger than $8\pi$ under a specific assumption associated with a Lyapunov functional. This finding suggests the relationship between solutions blowing up in infinite time and an unboundedness of a Lyapunov functional, which contrasts with the Keller--Segel system.