Boundedness of solutions in a two-species chemotaxis system

Chang-Jian Wang; Yuan-Hao Zang; Chang-Jian Wang; Yuan-Hao Zang

doi:10.3934/era.2025126

Electronic Research Archive

2025, Volume 33, Issue 5: 2862-2880. doi: 10.3934/era.2025126

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Research article

Boundedness of solutions in a two-species chemotaxis system

Chang-Jian Wang ^,,
Yuan-Hao Zang

School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, China

Received: 17 December 2024 Revised: 03 April 2025 Accepted: 18 April 2025 Published: 13 May 2025

In this paper, we consider an initial-Neumann boundary value problem for a two-species chemotaxis system
$ \begin{equation*} \left\{ \begin{array}{ll} \frac{\partial u}{\partial t} = \Delta u-\chi\nabla \cdot (u\nabla w)+u(a_{1}-b_{1}u^{m-1}+c_{1}v),\ &\ \ (x,t) \in \Omega\times(0,T_{\max}), \\[2.5mm] \frac{\partial v}{\partial t} = \Delta v-\xi\nabla \cdot (v\nabla w)+v(a_{2}-b_{2}v^{l-1}-c_{2}u), \ &\ \ (x,t) \in \Omega\times(0,T_{\max}), \\[2.5mm] \frac{\partial w}{\partial t} = \Delta w-(u^{\alpha}+v^{\beta})w, \ &\ \ (x,t) \in \Omega\times(0,T_{\max}), \end{array} \right. \end{equation*} $
where the domain $ \Omega\subset\mathbb{R}^{n}(n\geq 2) $ is bounded and smooth, $ T_{\max}\in (0, \infty], $ and parameters $ a_{i}, b_{i}, c_{i}, m, l, \alpha, $ $ \beta, \chi, \xi > 0 $ with $ m, l > 1, i = 1, 2. $ In the current work, we provide a sufficient condition of global classical solvability to the above system. More precisely, for some suitable initial data, if $ m > \max\{\frac{\alpha(n+2)}{2}, 1 \} $ and $ l > \max\{\frac{\beta(n+2)}{2}, 1 \}, $ then the system has a global classical solution. Compared to previous work, the existence result established here is more generalized, depending only on the nonlinear power exponents and spatial dimensions.
- predator-prey model,
- classical solutions,
- global existence,
- nonlinear consumption
Citation: Chang-Jian Wang, Yuan-Hao Zang. Boundedness of solutions in a two-species chemotaxis system[J]. Electronic Research Archive, 2025, 33(5): 2862-2880. doi: 10.3934/era.2025126

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Abstract

In this paper, we consider an initial-Neumann boundary value problem for a two-species chemotaxis system

$ \begin{equation*} \left\{ \begin{array}{ll} \frac{\partial u}{\partial t} = \Delta u-\chi\nabla \cdot (u\nabla w)+u(a_{1}-b_{1}u^{m-1}+c_{1}v),\ &\ \ (x,t) \in \Omega\times(0,T_{\max}), \\[2.5mm] \frac{\partial v}{\partial t} = \Delta v-\xi\nabla \cdot (v\nabla w)+v(a_{2}-b_{2}v^{l-1}-c_{2}u), \ &\ \ (x,t) \in \Omega\times(0,T_{\max}), \\[2.5mm] \frac{\partial w}{\partial t} = \Delta w-(u^{\alpha}+v^{\beta})w, \ &\ \ (x,t) \in \Omega\times(0,T_{\max}), \end{array} \right. \end{equation*} $

where the domain $ \Omega\subset\mathbb{R}^{n}(n\geq 2) $ is bounded and smooth, $ T_{\max}\in (0, \infty], $ and parameters $ a_{i}, b_{i}, c_{i}, m, l, \alpha, $ $ \beta, \chi, \xi > 0 $ with $ m, l > 1, i = 1, 2. $ In the current work, we provide a sufficient condition of global classical solvability to the above system. More precisely, for some suitable initial data, if $ m > \max\{\frac{\alpha(n+2)}{2}, 1 \} $ and $ l > \max\{\frac{\beta(n+2)}{2}, 1 \}, $ then the system has a global classical solution. Compared to previous work, the existence result established here is more generalized, depending only on the nonlinear power exponents and spatial dimensions.

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