Global existence and boundedness to a two-species chemotaxis-competition model with singular sensitivity

In the present study, we investigate the chemotaxis-consumption system of two competing species which are attracted by the same signal substance $$\begin{aligned} \left\{ \begin{array}{llll} u_t=\Delta u-\chi _1\nabla \cdot (\frac{u}{w}\nabla w)+u(a_1-b_1u-c_1v),&{}\quad x\in \Omega ,\quad t>0,\\ v_t=\Delta v-\chi _2\nabla \cdot (\frac{v}{w}\nabla w)+v(a_2-b_2v-c_2u),&{}\quad x\in \Omega ,\quad t>0,\\ w_t=\Delta w-(\alpha u+\beta v)w,&{}\quad x\in \Omega ,\quad t>0, \end{array} \right. \end{aligned}$$ 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 $$\min \{\frac{b_1}{3\alpha +\beta }, \frac{b_2}{\alpha +3\beta }\}>\frac{n-2}{4n}$$ .