Journal of Evolution Equations ( IF 1.4 ) Pub Date : 2020-11-21 , DOI: 10.1007/s00028-020-00650-6 Xinyu Tu , Chun-Lei Tang , Shuyan Qiu
We study herein the initial boundary value problem for a two-species chemotaxis competition system with loop
$$\begin{aligned} \left\{ \begin{array}{ll} \partial _{t} u_{1}=\Delta u_{1}-\chi _{11}\nabla \cdot (u_{1}\nabla v_{1}) -\chi _{12}\nabla \cdot (u_{1}\nabla v_{2}) +\mu _{1}u_{1}(1-u_{1}-a_{1}u_{2}),&{}\quad x\in \Omega ,\quad t>0,\\ \partial _{t} u_{2}=\Delta u_{2}-\chi _{21}\nabla \cdot (u_{2}\nabla v_{1}) -\chi _{22}\nabla \cdot (u_{2}\nabla v_{2}) +\mu _{2}u_{2}(1-u_{2}-a_{2}u_{1}), &{}\quad x\in \Omega ,\quad t>0,\\ \partial _t v_1=\Delta v_{1}- v_{1}+u_{1}+u_{2}, &{}\quad x\in \Omega ,\quad t>0,\\ \partial _t v_2=\Delta v_{2}- v_{2}+u_{1}+u_{2}, &{}\quad x\in \Omega ,\quad t>0,\\ \end{array}\right. \end{aligned}$$under the homogeneous Neumann boundary condition, where \(\Omega \subset {\mathbb {R}}^{n} (n\ge 3)\) is a smooth and bounded domain, \(\chi _{ij}>0\), \(\mu _{i}>0\), \(a_i>0\) \((i, j=1, 2)\). In the radial symmetric setting, for any \(T>0\) and \(L>0\), it is proved that there exists positive initial data such that the corresponding solution \((u_1, u_2, v_1, v_2)\) satisfies
$$\begin{aligned} u_1(x_{L}, t_{L})>L\quad \text {or}\quad u_2(x_{L}, t_{L})>L\quad \text {for some}\quad (x_{L}, t_{L})\in \Omega \times (0, T). \end{aligned}$$Moreover, when \(\chi _{11}=\chi _{12}, \chi _{21}=\chi _{22}\), \(\mu =\max \{\mu _1, \mu _2\}\in (0, 1)\), one can find initial data \( (u_{10}, u_{20}, v_{10}, v_{20})\in \left( C^0({\overline{\Omega }})\right) ^2\times \left( W^{1, \infty }(\Omega )\right) ^2\), which is irrelevant to \(\mu \), such that for all \(\mu \in (0, 1)\), the corresponding solution \((u_{1, \mu }, u_{2, \mu }, v_{1, \mu }, v_{2, \mu })\) fulfills
$$\begin{aligned} u_{1, \mu }(x_{\mu }, t_{\mu })>\frac{L}{\mu }\quad \text {or}\quad u_{2, \mu }(x_{\mu }, t_{\mu })>\frac{L}{\mu }\quad \text {for some}\quad (x_{\mu }, t_{\mu })\in \Omega \times (0, T). \end{aligned}$$In particular, it is proved that blowup for one of the chemotactic species implies also blowup for the other one at the same time.
中文翻译：
具有循环的趋化竞争系统中人口密度大的现象
我们在这里研究带环的两种种群趋化竞争系统的初始边值问题
$$ \ begin {aligned} \ left \ {\ begin {array} {ll} \ partial _ {t} u_ {1} = \ Delta u_ {1}-\ chi _ {11} \ nabla \ cdot（u_ { 1} \ nabla v_ {1}）-\ chi _ {12} \ nabla \ cdot（u_ {1} \ nabla v_ {2}）+ \ mu _ {1} u_ {1}（1-u_ {1} -a_ {1} u_ {2}），＆{} \ quad x \ in \ Omega，\ quad t> 0，\\ \ partial _ {t} u_ {2} = \ Delta u_ {2}-\ chi _ {21} \ nabla \ cdot（u_ {2} \ nabla v_ {1}）-\ chi _ {22} \ nabla \ cdot（u_ {2} \ nabla v_ {2}）+ \ mu _ {2} u_ {2}（1-u_ {2} -a_ {2} u_ {1}），＆{} \ quad x \ in \ Omega，\ quad t> 0，\\ \ partial _t v_1 = \ Delta v_ { 1}-v_ {1} + u_ {1} + u_ {2}，＆{} \ quad x \ in \ Omega，\ quad t> 0，\\ \ partial _t v_2 = \ Delta v_ {2}-v_ {2} + u_ {1} + u_ {2}，＆{} \ quad x \在\ Omega中，\ quad t> 0，\\ \ end {array} \ right。\ end {aligned} $$在齐次Neumann边界条件下，其中\（\ Omega \ subset {\ mathbb {R}} ^ {n}（n \ ge 3）\）是一个光滑且有界的域，\（\ chi _ {ij}> 0 \），\（\ mu _ {i}> 0 \），\（a_i> 0 \） \（（i，j = 1，2）\）。在径向对称设置中，对于任何\（T> 0 \）和\（L> 0 \），都证明存在正的初始数据，使得对应的解\（（u_1，u_2，v_1，v_2）\ ）满足
$$ \ begin {aligned} u_1（x_ {L}，t_ {L}）> L \ quad \ text {or} \ quad u_2（x_ {L}，t_ {L}）> L \ quad \ text {for } \ quad（x_ {L}，t_ {L}）\ in \ Omega \ times（0，T）。\ end {aligned} $$此外，当\（\ chi _ {11} = \ chi _ {12}，\ chi _ {21} = \ chi _ {22} \）时，\（\ mu = \ max \ {\ mu _1，\ mu _2 \} \ in（0，1）\）中，一个人可以找到\ left（C ^ 0（\ u {{10}，u_ {20}，v_ {10}，v_ {20}）\ {\ overline {\ Omega}}）\ right）^ 2 \ times \ left（W ^ {1，\ infty}（\ Omega} \ right）^ 2 \），与\（\ mu \）无关，这样对于所有\（\ mu \ in（0，1）\），对应的解决方案\（（u_ {1，\ mu}，u_ {2，\ mu}，v_ {1，\ mu}，v_ { 2，\ mu}）\）满足
$$ \ begin {aligned} u_ {1，\ mu}（x _ {\ mu}，t _ {\ mu}）> \ frac {L} {\ mu} \ quad \ text {or} \ quad u_ {2， \ mu}（x _ {\ mu}，t _ {\ mu}）> \ frac {L} {\ mu} \ quad \ text {for some} \ quad（x _ {\\ mu}，t _ {\ mu}）\在\ Omega \ times（0，T）中。\ end {aligned} $$特别地，已证明一种趋化性物质的爆炸也意味着同时另一种趋化性物质的爆炸。
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