Global boundedness for a two-dimensional doubly degenerate nutrient taxis system

2024/11/02 by Zhiguang Zhang, Yuxiang Li, Zhang, Zhiguang +1 · 1 citation
Mathematics · Medicine · #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2411.01133

openalex publication_date 2024/11/02 · openalex created_date 2024/11/14 · openalex updated_date 2026/07/28

Abstract

This paper is concerned with the doubly degenerate nutrient taxis system u_t=∇ ⋅(u^l-1 v ∇ u)- ∇ ⋅(u^l v ∇ v)+ uv and v_t=Δv-u v for some l \geqslant 1, subjected to homogeneous Neumann boundary conditions in a smooth bounded convex domain Ω⊂ ℝⁿ (n \leqslant 2). Through distinct approaches, we establish that for sufficiently regular initial data, in two-dimensional contexts, if l ∈[1,3], then the system possesses global weak solutions, and in one-dimensional settings, the same conclusion holds for l ∈[1,∞). Notably, the solution remains uniformly bounded when l ∈[1,∞) in one dimension or l ∈(1,3] in two dimensions.

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