Concentration phenomena to a chemotaxis system with indirect signal production

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.