Stabilization of arbitrary structures in a three-dimensional doubly degenerate nutrient taxis system
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| Format: | Preprint |
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2026
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| author | De-Ji-Xiang-Mao Huang, Ai Wang, Yifu |
| author_facet | De-Ji-Xiang-Mao Huang, Ai Wang, Yifu |
| contents | The doubly degenerate nutrient taxis system \begin{equation}\label {0.1} \left\{ \begin{aligned} &u_{t}=\nabla \cdot (uv\nabla u)-χ\nabla \cdot (u^αv\nabla v)+\ell uv,&x\in Ω,\, t>0,\\ & v_{t}=Δv-uv,&x\in Ω,\, t>0,\\ \end{aligned} \right. \end{equation} is considered under zero-flux boundary conditions in a smoothly bounded domain $Ω\subset\mathbb{R}^3$ where $α>0,χ>0$ and $\ell> 0$. By developing a novel class of functional inequalities to address the challenges posed by
the doubly degenerate diffusion mechanism in \eqref{0.1}, it is shown that for $α\in(\frac{3}{2},\frac{19}{12})$, the associated initial-boundary value problem admits a global continuous weak solution for sufficiently regular initial data. Furthermore, in an appropriate topological setting, this solution converges to an equilibrium $(u_\infty, 0)$ as $t\rightarrow \infty$. Notably, the limiting profile $u_{\infty}$ is non-homogeneous when the initial signal concentration $v_0$ is sufficiently small, provided the initial data $u_0$ is not identically constant. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_12218 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stabilization of arbitrary structures in a three-dimensional doubly degenerate nutrient taxis system De-Ji-Xiang-Mao Huang, Ai Wang, Yifu Analysis of PDEs 35K59, 35K65, 35B40, 92C17 The doubly degenerate nutrient taxis system \begin{equation}\label {0.1} \left\{ \begin{aligned} &u_{t}=\nabla \cdot (uv\nabla u)-χ\nabla \cdot (u^αv\nabla v)+\ell uv,&x\in Ω,\, t>0,\\ & v_{t}=Δv-uv,&x\in Ω,\, t>0,\\ \end{aligned} \right. \end{equation} is considered under zero-flux boundary conditions in a smoothly bounded domain $Ω\subset\mathbb{R}^3$ where $α>0,χ>0$ and $\ell> 0$. By developing a novel class of functional inequalities to address the challenges posed by the doubly degenerate diffusion mechanism in \eqref{0.1}, it is shown that for $α\in(\frac{3}{2},\frac{19}{12})$, the associated initial-boundary value problem admits a global continuous weak solution for sufficiently regular initial data. Furthermore, in an appropriate topological setting, this solution converges to an equilibrium $(u_\infty, 0)$ as $t\rightarrow \infty$. Notably, the limiting profile $u_{\infty}$ is non-homogeneous when the initial signal concentration $v_0$ is sufficiently small, provided the initial data $u_0$ is not identically constant. |
| title | Stabilization of arbitrary structures in a three-dimensional doubly degenerate nutrient taxis system |
| topic | Analysis of PDEs 35K59, 35K65, 35B40, 92C17 |
| url | https://arxiv.org/abs/2601.12218 |