Large-time behavior in an attraction--repulsion chemotaxis system
Abstract
We investigate the large-time behavior of small perturbations of stable constant steady states for an attraction-repulsion chemotaxis system in n-dimensional Euclidean space. We consider integrable perturbations in one space dimension and, in higher dimensions, perturbations belonging to Lebesgue spaces with suitable exponents. We first establish decay estimates throughout the full admissible range of Lebesgue exponents and show that the nonlinear perturbation is asymptotically approximated by the corresponding linearized evolution. We then identify the leading term of the linearized evolution as an effective heat flow whose diffusion coefficient is explicitly determined by the parameters of the system. Consequently, when the integrability exponent is strictly smaller than the space dimension, the leading asymptotic profile is governed by the heat equation. At the critical endpoint, the difference between the nonlinear perturbation and the effective heat flow vanishes under the natural parabolic scaling. In the one-dimensional integrable case, this yields a Gaussian profile determined by the total mass of the initial perturbation.
Related
- Emergence of large densities in a chemotaxis system with signal-dependent motility
- Generalized Determinantal Point Processes for Attraction and Repulsion with Distinct Interaction Kernels
- The nonlocal attraction-repulsion transport equation with power kernels
- Global Boundedness and Finite-Time Blow-up for a Quasilinear Chemotaxis--May--Nowak System
- Sharp blow-up condition for a two-species cross attraction system in 3 or more dimensions
BibTeX
@misc{wakui2026largetime,
title = {{Large-time behavior in an attraction--repulsion chemotaxis system}},
author = {Hiroshi Wakui and Tetsuya Yamada},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18258},
eprint = {2609.18258},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18258},
}
RIS
TY - UNPB
TI - Large-time behavior in an attraction--repulsion chemotaxis system
AU - Hiroshi Wakui
AU - Tetsuya Yamada
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18258
UR - https://arxiv.org/abs/2609.18258
AB - We investigate the large-time behavior of small perturbations of stable constant steady states for an attraction-repulsion chemotaxis system in n-dimensional Euclidean space. We consider integrable perturbations in one space dimension and, in higher dimensions, perturbations belonging to Lebesgue spaces with suitable exponents. We first establish decay estimates throughout the full admissible range of Lebesgue exponents and show that the nonlinear perturbation is asymptotically approximated by the corresponding linearized evolution. We then identify the leading term of the linearized evolution as an effective heat flow whose diffusion coefficient is explicitly determined by the parameters of the system. Consequently, when the integrability exponent is strictly smaller than the space dimension, the leading asymptotic profile is governed by the heat equation. At the critical endpoint, the difference between the nonlinear perturbation and the effective heat flow vanishes under the natural parabolic scaling. In the one-dimensional integrable case, this yields a Gaussian profile determined by the total mass of the initial perturbation.
ER -