journal article

Journal Article

Text

Journal of Differential Equations. 2026, 450, p. 113745

Existence of axis-symmetric blow-up solution with multiple peak aggregations for the 2-D Keller-Segel systems coupled bipolar source and sink flow

Academic Press Inc.

English

Text

2026-01-05

https://doi.org/10.1016/j.jde.2025.113745

Seki Y., Shibata K., Sugiyama Y.. Existence of axis-symmetric blow-up solution with multiple peak aggregations for the 2-D Keller-Segel systems coupled bipolar source and sink flow. Journal of Differential Equations 450, 113745 (2026); https://doi.org/10.1016/j.jde.2025.113745.

We consider the Keller-Segel systems in Rd, coupled with a bipolar source and sink flow. Focusing on the two-dimensional case (d=2), we establish finite-time blow-up of solutions under an axis-symmetric setting, without requiring the solutions to be radial. In particular, we prove that multiple blow-up points appear in pairs (i.e., in even numbers) away from the origin, lying on the x1-axis and exhibiting axis-symmetry about the x2-axis. This result holds for initial data with total mass strictly greater than 16π, and stands in contrast to the classical radial setting, where blow-up is confined to the origin. A crucial part of our analysis is a sharp ε-regularity theorem, originally developed for the classical Keller-Segel systems and first established by Luckhaus–Sugiyama–Velázquez [12]. This theorem states that if the local mass around x1 is sufficiently small at some time t1, then the solution remains locally bounded in a suitable parabolic cylinder in space–time centered at (x1,t1). Compared to the classical ε-regularity theorem, it requires weaker assumptions and yields weaker conclusions, making it a form of partial regularity that is particularly essential for analyzing blow-up singularities. Based on this sharp ε-regularity theorem, we further prove that only finitely many blow-up points appear as singular sets, and the asymptotic profile is characterized as the sum of a finite number of δ-functions and a regular part in L1(R2). Moreover, our results reveal that multi-peak blow-up phenomena can occur with or without the presence of non-trivial flow, highlighting the intricate interplay between diffusion, chemotaxis, and persistent advection. By accounting for non-decaying flow and employing precise blow-up criteria, we establish that the blow-up time can be bounded above by any prescribed threshold. These findings are justified through the construction of a time-local existence and extension theory for strong solutions, which incorporates both advection and mass conservation.

This article is licensed under a Creative Commons Attribution 4.0 International License.

102877

Existence of axis-symmetric blow-up solution with multiple peak aggregations for the 2-D Keller-Segel systems coupled bipolar source and sink flow

Academic Press Inc.

English

2026-01-05

Journal of Differential Equations

450

113745

https://doi.org/10.1016/j.jde.2025.113745

https://hdl.handle.net/11094/102877