Spatial inhomogeneity for a three-dimensional doubly degenerate nutrient system with indirect consumption
Abstract
This paper investigates the global dynamics of a doubly degenerate nutrient-taxis system with indirect consumption: { &ut=∇ · (uv∇ u)−∇ · (u2v∇ v)+ℓ vw,&x∈ Ω, t>0, & vt=Δv−vw,&x∈ Ω, t>0, &wt=Δw−w+u,&x∈Ω,t>0 . posed on a smooth bounded domain Ω⊂ℝ3 with no-flux boundary conditions. It is shown that for suitably regular initial data (u0,v0,w0), the associated initial-boundary value problem admits a global weak solution. Furthermore, in an appropriate topological setting, this solution converges to an equilibrium (u∞, 0,w∞) as t→ ∞. Notably, when u0 is nonconstant and the mass of v0 is sufficiently small, the limiting profiles u∞ and w∞ are are spatially nonhomogeneous, capturing emergent patterning in nutrient-depleted environments. A cornerstone of our analysis is the introduction of novel functional inequalities, which provide estimates from below for the integral ∫Ωukv|∇ u|2 with some k>−1.
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BibTeX
@misc{huang2026spatial,
title = {{Spatial inhomogeneity for a three-dimensional doubly degenerate nutrient system with indirect consumption}},
author = {Ai Huang and Xiangmao De-ji and Jing Li and Yifu Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02607},
eprint = {2609.02607},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02607},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Spatial inhomogeneity for a three-dimensional doubly degenerate nutrient system with indirect consumption
AU - Ai Huang
AU - Xiangmao De-ji
AU - Jing Li
AU - Yifu Wang
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02607
UR - https://arxiv.org/abs/2609.02607
AB - This paper investigates the global dynamics of a doubly degenerate nutrient-taxis system with indirect consumption: { &u_t=∇ · (uv∇ u)−∇ · (u^2v∇ v)+ℓ vw,&x∈ Ω, t>0, & v_t=Δv−vw,&x∈ Ω, t>0, &w_t=Δw−w+u,&x∈Ω,t>0 . posed on a smooth bounded domain Ω⊂ℝ^3 with no-flux boundary conditions. It is shown that for suitably regular initial data (u_0,v_0,w_0), the associated initial-boundary value problem admits a global weak solution. Furthermore, in an appropriate topological setting, this solution converges to an equilibrium (u_∞, 0,w_∞) as t→ ∞. Notably, when u_0 is nonconstant and the mass of v_0 is sufficiently small, the limiting profiles u_∞ and w_∞ are are spatially nonhomogeneous, capturing emergent patterning in nutrient-depleted environments. A cornerstone of our analysis is the introduction of novel functional inequalities, which provide estimates from below for the integral ∫_Ωu^kv|∇ u|^2 with some k>−1.
ER -