Global solvability for doubly degenerate nutrient taxis system with a wide range of bacterial responses in physical dimension

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

paper · doi:10.48550/arxiv.2508.03268

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∈ℝ is the bacterial response parameter. Global solvability of weak solutions to this taxis system is highly challenging due to not only the doubly nonlinear diffusion and its degeneracy but also the strong chemotactic effect, where the latter is strong at the large species density if α is close to 2. 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=2 with initial data of small size if α=2; \item In [Z. Zhang and Y. Li, arXiv:2405.20637, 2024] for α=2, N=2; and \item In [G. Li, J. Differ. Equ., 2022] for (7)/(6)<α<(13)/(9), 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\n cross-diffusion as the parabolic limit of a velocity-jump process
• Locally bounded global solutions to a chemotaxis consumption model with\n singular sensitivity and nonlinear diffusion
• The effects of nutrient chemotaxis on bacterial aggregation patterns\n 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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