Global boundedness of the chemotaxis system with weakly singular sensitivity and nonlocal term in any dimension
Abstract
This paper is concerned with the parabolic-elliptic chemotaxis system involving weakly singular sensitivity and a nonlocal source: ut=Δu−χ∇·(u/vk∇ v) +uα(r−μ∫Ωuβ dx) and 0=Δv−v+uγ under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂ℝN with N≥1, where χ,r,μ,γ>0, k∈(0,1) and α,β≥1. In view of the enhanced aggregation induced by singular sensitivity and the suppression of excessive population growth by the nonlocal damping, we identify sufficient conditions under which solutions remain globally bounded. We show that classical solutions are globally bounded in the following cases: \par\smallskip \begin{center} \small \setlength{\tabcolsep}{5pt} \setlength{\arrayrulewidth}{0.3pt} \renewcommand{\arraystretch}{1.25} \begin{tabular}{c|c|c|c} \multirow{2}{*}{β>1} & \multirow{2}{*}{γ=1} & N=1 & 1≤α<1+2β \\ \cline{3-4} & & N≥2 & 1≤α<2, α+β>2+N/2, [−0.5mm] or 2≤α<1+2β/N \\ \hline \multirow{2}{*}{β=1} & γ=1 & N=1 & \multirow{2}{*}{ l@ l@ l Case 1: & 1≤α<1+2/N & if m0<r/μ, [4pt] Case 2: & α≥1 & if m0≥r/μ } \\ \cline{2-3} & 0<γ<2/N & N≥2 & \\ \end{tabular} \end{center} \par \smallskip \noindent Here m0:=∫Ωu0 denotes the initial total mass. It is worth noting that both the uniform-in-time Lp-estimate and the L∞-bound obtained by Moser iteration are derived without first establishing a uniform positive lower bound for v.
Related
- Boundedness in a chemotaxis system with nonlinear diffusion and singular sensitivity
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- A SHARP MASS THRESHOLD FOR BOUNDEDNESS IN A CRITICAL QUASILINEAR CHEMOTAXIS MODEL WITH INDIRECT SIGNAL PRODUCTION: THE RADIALLY SYMMETRIC CASE
- Improved Global Boundedness Criteria via Refined Bootstrap Estimates for a Quasilinear Chemotaxis-May-Nowak System
- Global boundedness in a Chemotaxis-May-Nowak model for virus dynamics with logistic damping
BibTeX
@misc{cao2026global,
title = {{Global boundedness of the chemotaxis system with weakly singular sensitivity and nonlocal term in any dimension}},
author = {Yang Cao and Qingchun Li and Jing Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14443},
eprint = {2609.14443},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14443},
}
RIS
TY - UNPB
TI - Global boundedness of the chemotaxis system with weakly singular sensitivity and nonlocal term in any dimension
AU - Yang Cao
AU - Qingchun Li
AU - Jing Zhang
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14443
UR - https://arxiv.org/abs/2609.14443
AB - This paper is concerned with the parabolic-elliptic chemotaxis system involving weakly singular sensitivity and a nonlocal source: u_t=Δu−χ∇·(u/v^k∇ v) +u^α(r−μ∫_Ωu^β dx) and 0=Δv−v+u^γ under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂ℝ^N with N≥1, where χ,r,μ,γ>0, k∈(0,1) and α,β≥1. In view of the enhanced aggregation induced by singular sensitivity and the suppression of excessive population growth by the nonlocal damping, we identify sufficient conditions under which solutions remain globally bounded. We show that classical solutions are globally bounded in the following cases: \par\smallskip \begin{center} \small \setlength{\tabcolsep}{5pt} \setlength{\arrayrulewidth}{0.3pt} \renewcommand{\arraystretch}{1.25} \begin{tabular}{c|c|c|c} \multirow{2}{*}{β>1} & \multirow{2}{*}{γ=1} & N=1 & 1≤α<1+2β \\ \cline{3-4} & & N≥2 & 1≤α<2, α+β>2+N/2, [−0.5mm] or 2≤α<1+2β/N \\ \hline \multirow{2}{*}{β=1} & γ=1 & N=1 & \multirow{2}{*}{ l@ l@ l Case 1: & 1≤α<1+2/N & if m_0<r/μ, [4pt] Case 2: & α≥1 & if m_0≥r/μ } \\ \cline{2-3} & 0<γ<2/N & N≥2 & \\ \end{tabular} \end{center} \par \smallskip \noindent Here m_0:=∫_Ωu_0 denotes the initial total mass. It is worth noting that both the uniform-in-time L^p-estimate and the L^∞-bound obtained by Moser iteration are derived without first establishing a uniform positive lower bound for v.
ER -