Boundedness and exponential stabilization in a degenerate migration-consumption system with logistic damping

Ai Huang; Jing Li; Yifu Wang

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arXiv · 2610.16399

Boundedness and exponential stabilization in a degenerate migration-consumption system with logistic damping

Ai Huang·Jing Li·Yifu Wang

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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 $α\ge1$. We prove uniform boundedness in $L^\infty(Ω)\times W^{1,\infty}(Ω)$ 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.

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BibTeX RIS

Ai Huang, Jing Li, Yifu Wang. 2026-10-12. Boundedness and exponential stabilization in a degenerate migration-consumption system with logistic damping. https://arxiv.org/abs/2610.16399

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