Existence of pulse solutions in a bulk--surface reaction--diffusion system for cell polarization
Abstract
In this paper, we prove the existence of stationary pulse solutions for a bulk-surface reaction-diffusion system modeling cell polarization in two-dimensional domains. Using a singular perturbation method, we construct pulse solutions whose centers lie near non-degenerate critical points of a geometry-dependent potential expressed in terms of the Neumann Green's function, as formally predicted in \cite{watanabe2026}. The main analytical difficulty arises from the half-Laplacian induced by the logarithmic singularity of the Green's function, resulting in algebraically rather than exponentially decaying inner solutions. Our analysis may also serve as a useful guide for applying singular perturbation methods to other bulk-surface type systems.
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BibTeX
@misc{watanabe2026existence,
title = {{Existence of pulse solutions in a bulk--surface reaction--diffusion system for cell polarization}},
author = {Riku Watanabe},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11512},
eprint = {2610.11512},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11512},
}
RIS
TY - UNPB
TI - Existence of pulse solutions in a bulk--surface reaction--diffusion system for cell polarization
AU - Riku Watanabe
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11512
UR - https://arxiv.org/abs/2610.11512
AB - In this paper, we prove the existence of stationary pulse solutions for a bulk-surface reaction-diffusion system modeling cell polarization in two-dimensional domains. Using a singular perturbation method, we construct pulse solutions whose centers lie near non-degenerate critical points of a geometry-dependent potential expressed in terms of the Neumann Green's function, as formally predicted in \cite{watanabe2026}. The main analytical difficulty arises from the half-Laplacian induced by the logarithmic singularity of the Green's function, resulting in algebraically rather than exponentially decaying inner solutions. Our analysis may also serve as a useful guide for applying singular perturbation methods to other bulk-surface type systems.
ER -