In the present study, we investigate the chemotaxis-consumption system of two competing species which are attracted by the same signal substance

associated with homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega \subset R^{n}(n\ge 1)\), where the parameters \(\alpha \), \(\beta \), \(\chi _i\), \(a_i\), \(b_i\), \(c_i\), \(i=1, 2\) are supposed to be positive. When \(n=1\), it is shown that whenever the initial data \((u_0, v_0, w_0)\) are positive and suitably regular, the associated initial-boundary value problem admits a globally defined bounded classical solution for any \(\chi _i\), \(b_i>0\, (i =1,2)\). When \(n=2\), we establish that if \(\max \{\chi _1, \chi _2\}<1\), then the global solution exists regardless of the sizes of \(b_1>0\) and \(b_2>0\), or if \(\min \{\chi _1, \chi _2\}\ge 1\), then there are \(b^*_i(\chi _i)\,(i =1,2)>0\) such that the global classical solution also exists when \(b_i>b^*_i(\chi _i)\,(i =1,2)\) . Moreover, the global boundedness of the classical solution is determined as well, that is, there exist \(\lambda _i(\Omega )>0\) and \(\gamma _i(\Omega )>0\) such that the global solution (u, v, w) is uniformly bounded in time provided that \(b_i>\lambda _i(\Omega )a_i+\gamma _i(\Omega )\) for \(\max \{\chi _1, \chi _2\}<1\) or \(b_i>\{b^*_i(\chi _i), \lambda _i(\Omega )a_i+\gamma _i(\Omega )\}\) for \(\min \{\chi _1, \chi _2\}\ge 1\) with \(i=1,2\), respectively. Furthermore, when \(n\ge 3\), the corresponding initial-boundary value problem possesses a unique global classical solution under the conditions that \(\max \{\chi _1, \chi _2\}<\sqrt{\frac{2}{n}}\) and \(\min \{\frac{b_1}{3\alpha +\beta }, \frac{b_2}{\alpha +3\beta }\}>\frac{n-2}{4n}\).

在本研究中，我们研究了两个竞争物种被同一信号物质吸引的趋化-消耗系统

与光滑有界域\（\ Omega \ subset R ^ {n}（n \ ge 1）\）中的齐次Neumann边界条件相关，其中参数\（\ alpha \），\（\ beta \），\（ \ chi _i \），\（a_i \），\（b_i \），\（c_i \），\（i = 1，2 \）应该是正数。当\（n = 1 \）时，表明只要初始数据\（（u_0，v_0，w_0）\）为正且适当地规则，相关联的初始边界值问题就允许对任何\（\ chi _i \），\（b_i> 0 \，（i = 1,2）\）。当\（n = 2 \），我们确定如果\（\ max \ {\ chi _1，\ chi _2 \} <1 \），则存在全局解决方案，而与\（b_1> 0 \）和\（b_2> 0 \）的大小无关，或者\\\ min \ {\ chi _1，\ chi _2 \} \ ge 1 \），则存在\（b ^ * _ i（\ chi _i）\，（i = 1,2）> 0 \ ），这样当\（b_i> b ^ * _ i（\ chi _i）\，（i = 1,2）\）时也存在全局经典解。此外，经典溶液的全球有界被确定为好，即，存在\（\拉姆达_i（\欧米茄）> 0 \）和\（\伽马_i（\欧米茄）> 0 \） ，使得所述全局解（u，  v，  w）在时间上是一致的，只要\（b_i> \ lambda _i（\ Omega）a_i + \ gamma _i（\ Omega）\）表示\（\ max \ {\ chi _1，\ chi _2 \} <1 \）或\（b_i> \ {b ^ * _i（\ chi _i），\ lambda _i（\ Omega）a_i + \ gamma _i（\ Omega} \} \）对于\（\ min \ {\ chi _1，\ chi _2 \} \ ge 1 \）与\ （i = 1,2 \）分别。此外，当\（n \ ge 3 \）时，对应的初始边界值问题在\（\ max \ {\ chi _1，\ chi _2 \} <\ sqrt {\ frac {2} {n}} \）和\（\ min \ {\ frac {b_1} {3 \ alpha + \ beta}，\ frac {b_2} {\ alpha +3 \ beta} \}> \ frac {n -2} {4n} \）。