Global weak solvability for doubly degenerate nutrient taxis system in physical dimension

2025/08/05 by Bao-Ngoc Tran, Juan Yang, Tran, Bao-Ngoc +1 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.2508.03268

arxiv created 2026/08/04 · arxiv updated 2026/08/06

Abstract

Motivated by the study of bacteria's response to environmental conditions, we consider the doubly degenerate nutrient taxis system \begincases u_t=∇⋅(uv∇ u)-χ∇⋅(u^αv∇ v)+ℓ uv,
v_t=Δv-uv, \endcases subjected to no-flux boundary conditions and smooth initial data, where α≥0 is the bacterial response parameter. Global solvability of weak solutions to this taxis system appears to be highly challenging due to the difficulty of quantifying the dissipation of the doubly degenerate diffusive flux, the strong chemotactic effect, particularly as α is close to 2, and the dimensionally dependent limitation, which is the most difficult one. Recent findings on the global weak solvability for the considered system are summarised as follows \beginitemize \item In [M. Winkler, Trans. Amer. Math. Soc., 2021] for α=2, N=1; \item In [M. Winkler, J. Differ. Equ., 2024] for 1≤α≤ 2, N=2with initial data of small size if α=2; \item In [Z. Zhang, Y. Li, Math. Models Methods Appl. Sci., 2026] for α=2, N=2; \item In [G. Li, J. Differ. Equ., 2022] for (7)/(6)<α<(13)/(9), N=3; and \item In [X.M. De-Ji, A. Huang, Y. Wang, Eur. J. Appl. Math.] for (3)/(2)<α<(19)/(12), N=3. \enditemize Our work aims to provide a picture of global weak solvability for 0≤ α<2 in the physically dimensional setting N=3. As suggested by the analysis, it is divided into three separable cases, including (i) 0≤α≤1: Weak chemotaxis effect; (ii) 1<α≤3/2: Moderate chemotaxis effect; and (iii) 3/2<α<2: Strong chemotaxis effect.

Citations

• Global weak solutions to a two-dimensional doubly degenerate nutrient taxis system with logistic source
• Boundedness in a two-dimensional doubly degenerate nutrient taxis system
• Rigorous fast signal diffusion limit and convergence rates with the initial layer effect in a competitive chemotaxis system
• Existence theory and qualitative analysis for a fully cross-diffusive predator-prey system
• Mathematical models for chemotaxis and their applications in self-organisation phenomena
• Derivation of a bacterial nutrient-taxis system with doubly degenerate cross-diffusion as the parabolic limit of a velocity-jump process
• Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion
• The effects of nutrient chemotaxis on bacterial aggregation patterns with non-linear degenerate cross diffusion
• Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity
• Model for chemotaxis

Cited by

• Stability and uniqueness of bounded weak solutions to triangular degenerate cross-diffusion systems

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• Boundedness in a two-dimensional doubly degenerate nutrient taxis system