Zeitschrift für angewandte Mathematik und Physik ( IF 1.7 ) Pub Date : 2020-10-13 , DOI: 10.1007/s00033-020-01413-6 Xinyu Tu , Chunlai Mu , Shuyan Qiu , Li Yang
This paper deals with the two-species chemotaxis-competition system with loop
$$\begin{aligned} \left\{ \begin{array}{llll} \partial _{t} u_{1}=d_1\Delta u_{1}-\chi _{11}\nabla \cdot (u_{1}\nabla v_{1}) -\chi _{12}\nabla \cdot (u_{1}\nabla v_{2}) +\mu _{1}u_{1}(1-u_{1}-a_{1}u_{2}),\\ \partial _{t} u_{2}=d_2\Delta u_{2}-\chi _{21}\nabla \cdot (u_{2}\nabla v_{1}) -\chi _{22}\nabla \cdot (u_{2}\nabla v_{2}) +\mu _{2}u_{2}(1-u_{2}-a_{2}u_{1}),\\ \partial _t v_1=d_3\Delta v_{1}-\lambda _{1} v_{1}+\alpha _{11}u_{1}+\alpha _{12}u_{2},\\ \partial _t v_2=d_4\Delta v_{2}-\lambda _{2} v_{2}+\alpha _{21}u_{1}+\alpha _{22}u_{2}, \\ \end{array} \right. \end{aligned}$$subject to homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega \subset {\mathbb {R}}^{3}\), where \(\chi _{ij}>0\), \(\mu _{i}>0\), \(a_i>0\), \(\alpha _{ij}>0\), \(\lambda _{i}>0\), \(d_k>0\) \((i, j=1, 2, k=1, 2, 3, 4)\). Our main purpose is to extend the global boundedness result to the 3D setting. To address this issue, based on a new coupled function, by selecting sufficiently large \(\mu _1\) and \(\mu _2\), we construct a Gronwall type inequality which directly renders the uniform boundedness of solutions.
中文翻译：
带环的高维全抛物线趋化竞争系统的有界性
本文研究带环的两种种群趋化竞争系统
$$ \开始{aligned} \左\ {\开始{array} {llll} \部分_ {t} u_ {1} = d_1 \ Delta u_ {1}-\ chi _ {11} \ nabla \ cdot（u_ {1} \ nabla v_ {1}）-\ chi _ {12} \ nabla \ cdot（u_ {1} \ nabla v_ {2}）+ \ mu _ {1} u_ {1}（1-u_ {1 } -a_ {1} u_ {2}），\\ \部分_ {t} u_ {2} = d_2 \ Delta u_ {2}-\ chi _ {21} \ nabla \ cdot（u_ {2} \ nabla v_ {1}）-\ chi _ {22} \ nabla \ cdot（u_ {2} \ nabla v_ {2}）+ \ mu _ {2} u_ {2}（1-u_ {2} -a_ {2 } u_ {1}），\\ \部分_t v_1 = d_3 \ Delta v_ {1}-\ lambda _ {1} v_ {1} + \ alpha _ {11} u_ {1} + \ alpha _ {12} u_ {2}，\\ \部分_t v_2 = d_4 \ Delta v_ {2}-\ lambda _ {2} v_ {2} + \ alpha _ {21} u_ {1} + \ alpha _ {22} u_ { 2}，\\ \结束{array} \对。\ end {aligned} $$服从光滑界域\（\ Omega \子集{\ mathbb {R}} ^ {3} \）中的齐次Neumann边界条件，其中\（\ chi _ {ij}> 0 \），\（\ mu _ {i}> 0 \），\（a_i> 0 \），\（\ alpha _ {ij}> 0 \），\（\ lambda _ {i}> 0 \），\（d_k> 0 \） \ （（i，j = 1，2，k = 1，2，3，4）\）。我们的主要目的是将全局有界结果扩展到3D设置。为了解决这个问题，基于新的耦合函数，通过选择足够大的\（\ mu _1 \）和\（\ mu _2 \），我们构造了一个Gronwall型不等式，它直接呈现了解决方案的一致有界性。
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