Boundedness and exponential stabilization in a degenerate migration-consumption system with logistic damping
Abstract
We study the no-flux problem for a taxis-type migration-consumption system with logistic damping in a bounded planar domain, where the motility function ϕ vanishes at zero and degenerates algebraically of order α≥1. We prove uniform boundedness in L∞(Ω)× W1,∞(Ω) for every fixed damping coefficient b>0, without any smallness assumption on the initial data, thus improving the previous L2-boundedness results. For α>1, we further prove that the solution converges exponentially to the spatially homogeneous state (a/b,0), with an explicit admissible rate. The proof combines a weighted Gagliardo--Nirenberg type inequality adapted to the degeneracy with the dissipative structure of the signal equation. Therefore logistic damping drives the system toward homogeneity, despite the loss of uniform elliptic regularization.
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BibTeX
@misc{huang2026boundedness,
title = {{Boundedness and exponential stabilization in a degenerate migration-consumption system with logistic damping}},
author = {Ai Huang and Jing Li and Yifu Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16399},
eprint = {2610.16399},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16399},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Boundedness and exponential stabilization in a degenerate migration-consumption system with logistic damping
AU - Ai Huang
AU - Jing Li
AU - Yifu Wang
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16399
UR - https://arxiv.org/abs/2610.16399
AB - We study the no-flux problem for a taxis-type migration-consumption system with logistic damping in a bounded planar domain, where the motility function ϕ vanishes at zero and degenerates algebraically of order α≥1. We prove uniform boundedness in L^∞(Ω)× W^1,∞(Ω) for every fixed damping coefficient b>0, without any smallness assumption on the initial data, thus improving the previous L^2-boundedness results. For α>1, we further prove that the solution converges exponentially to the spatially homogeneous state (a/b,0), with an explicit admissible rate. The proof combines a weighted Gagliardo--Nirenberg type inequality adapted to the degeneracy with the dissipative structure of the signal equation. Therefore logistic damping drives the system toward homogeneity, despite the loss of uniform elliptic regularization.
ER -