Concentration phenomena to a chemotaxis system with indirect signal production
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915347712442368 |
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| author | Soga, Yuri |
| author_facet | Soga, Yuri |
| contents | 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 $m \ge 8π$, and a nonnegative function $f \in L^1(Ω)$ such that \begin{align*} u(\cdot,t_k) \stackrel{*}{\rightharpoonup} m δ(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 $m = 8π$ or $m > 8π$, 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 $8π$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_13411 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Concentration phenomena to a chemotaxis system with indirect signal production Soga, Yuri Analysis of PDEs 35B44 (Primary) 35B45, 35K20, 35K55, 92D40 (Secondary) 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 $m \ge 8π$, and a nonnegative function $f \in L^1(Ω)$ such that \begin{align*} u(\cdot,t_k) \stackrel{*}{\rightharpoonup} m δ(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 $m = 8π$ or $m > 8π$, 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 $8π$ 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. |
| title | Concentration phenomena to a chemotaxis system with indirect signal production |
| topic | Analysis of PDEs 35B44 (Primary) 35B45, 35K20, 35K55, 92D40 (Secondary) |
| url | https://arxiv.org/abs/2502.13411 |