Zeitschrift für angewandte Mathematik und Physik ( IF 1.7 ) Pub Date : 2020-07-16 , DOI: 10.1007/s00033-020-01363-z Guangyu Xu
This paper deals with the solution of two-species chemotaxis system
$$\begin{aligned} \left\{ \begin{array}{llll} u_t=d_1\Delta u-\chi _1\nabla \cdot (u\nabla w)+\mu _{1}u(1-u-a_{1}v),&{}\quad x\in \Omega ,\quad t>0,\\ v_t=d_2\Delta v-\chi _2\nabla \cdot (v\nabla w)+\mu _{2}v(1-a_{2}u-v),&{}\quad x\in \Omega ,\quad t>0,\\ 0=d_3\Delta w-\gamma w+\alpha u+\beta v&{}\quad x\in \Omega ,\quad t>0 \end{array} \right. \end{aligned}$$(0.1)in a smooth bounded domain \(\Omega \subset {\mathbb {R}}^N, N\ge 1\). When \(d_1=d_2=0\), we first establish the local well-posedness of corresponding hyperbolic–hyperbolic–elliptic problem with the help of some compactness arguments and then obtain a blowup in finite time result for this problem. Using this blow-up conclusion, we further consider model (0.1) with small \(d_1, d_2>0\), and we then get that for any given \(M>0\) and \(T>0\), one can find suitable large, radially symmetric initial data and some appropriate parameters such that the corresponding classical solution of (0.1) satisfies
$$\begin{aligned} u(x,t)+v(x,t)>M, \end{aligned}$$with some \(x\in \Omega \) and \(t\in (0, T)\).
中文翻译：
具有两个物种的趋化系统的承载能力和N维竞争动力学
本文讨论了两种物种趋化系统的解决方案
$$ \ begin {aligned} \ left \ {\ begin {array} {llll} u_t = d_1 \ Delta u- \ chi _1 \ nabla \ cdot（u \ nabla w）+ \ mu _ {1} u（1- u-a_ {1} v），＆{} \ quad x \ in \ Omega，\ quad t> 0，\\ v_t = d_2 \ Delta v- \ chi _2 \ nabla \ cdot（v \ nabla w）+ \ mu _ {2} v（1-a_ {2} uv），＆{} \ quad x \ in \ Omega，\ quad t> 0，\\ 0 = d_3 \ Delta w- \ gamma w + \ alpha u + \ beta v＆{} \ quad x \ in \ Omega，\ quad t> 0 \ end {array} \ right。\ end {aligned} $$（0.1）在光滑有界域\（\ Omega \ subset {\ mathbb {R}} ^ N，N \ ge 1 \）中。当\（d_1 = d_2 = 0 \）时，我们首先借助一些紧性参数建立相应的双曲线-双曲线-椭圆型问题的局部适定性，然后针对该问题获得有限时间的爆破。使用这个爆炸性结论，我们进一步考虑具有小\（d_1，d_2> 0 \）的模型（0.1），然后对于任何给定的\（M> 0 \）和\（T> 0 \）都可以得到它。可以找到合适的，大的，径向对称的初始数据和一些合适的参数，从而使（0.1）的相应经典解满足
$$ \ begin {aligned} u（x，t）+ v（x，t）> M，\ end {aligned} $$与一些\（x \ in \ Omega \）和\（t \ in（0，T）\）。
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