Boundedness in a chemotaxis system with nonlinear diffusion and singular sensitivity
Abstract
This paper studies the following chemotaxis system with nonlinear diffusion and singular sensitivity ut = ∇ · ( (u+a)m−1 ∇ u ) − χ∇ · ( u (∇ v)/vk ), [4pt] vt = Δv − v + u, in a smoothly bounded domain Ω⊂ ℝn, n ≥ 2, subject to no-flux boundary conditions, where χ,k,a,m>0. For suitably regular initial data, we prove global existence and uniform-in-time boundedness of classical solutions if one of the following conditions holds: & n=2, k∈(0,∞), m>1; & n≥ 3, k∈(0,1/2), m>1+((1−k)(n−2))/n; & n≥ 3, k∈[1/2,1), m>(3n−2)/2n; & n≥ 3, k∈[1,∞), m>max{1,(3n−2)/(2n+2)}. The main contribution is the physically relevant three-dimensional case. In particular, for every χ>0, a>0, m>1 and k≥ 1, the system admits a global bounded classical solution. Thus, in dimension three, nonlinear diffusion of porous-medium type removes the smallness restriction on the chemotactic sensitivity χ that earlier results require in the case m=1 and k≥ 1.
Related
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BibTeX
@misc{le2026boundedness,
title = {{Boundedness in a chemotaxis system with nonlinear diffusion and singular sensitivity}},
author = {Minh Le},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39507},
eprint = {2610.39507},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39507},
}
RIS
TY - UNPB
TI - Boundedness in a chemotaxis system with nonlinear diffusion and singular sensitivity
AU - Minh Le
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39507
UR - https://arxiv.org/abs/2610.39507
AB - This paper studies the following chemotaxis system with nonlinear diffusion and singular sensitivity u_t = ∇ · ( (u+a)^m−1 ∇ u ) − χ∇ · ( u (∇ v)/v^k ), [4pt] v_t = Δv − v + u, in a smoothly bounded domain Ω⊂ ℝ^n, n ≥ 2, subject to no-flux boundary conditions, where χ,k,a,m>0. For suitably regular initial data, we prove global existence and uniform-in-time boundedness of classical solutions if one of the following conditions holds: & n=2, k∈(0,∞), m>1; & n≥ 3, k∈(0,1/2), m>1+((1−k)(n−2))/n; & n≥ 3, k∈[1/2,1), m>(3n−2)/2n; & n≥ 3, k∈[1,∞), m>max{1,(3n−2)/(2n+2)}. The main contribution is the physically relevant three-dimensional case. In particular, for every χ>0, a>0, m>1 and k≥ 1, the system admits a global bounded classical solution. Thus, in dimension three, nonlinear diffusion of porous-medium type removes the smallness restriction on the chemotactic sensitivity χ that earlier results require in the case m=1 and k≥ 1.
ER -