Title: Two-species chemotaxis-competition system with singular sensitivity: Global existence, boundedness, and persistence

Abstract: This paper is concerned with the following parabolic-parabolic-elliptic chemotaxis system with singular sensitivity and Lotka-Volterra competitive kinetics,$$ \begin{cases} u_t=\Delta u-\chi_1\nabla\cdot (\frac{u}{w}\nabla w)+u(a_1-b_1u-c_1v),\quad &x\in\Omega\cr v_t=\Delta v-\chi_2\nabla\cdot (\frac{v}{w}\nabla w)+v(a_2-b_2v-c_2u),\quad &x\in\Omega\cr 0=\Delta w-\mu w +\nu u+ \lambda v,\quad &x\in\Omega\cr \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=\frac{\partial w}{\partial n}=0,\quad &x\in\partial\Omega, \end{cases} $$ where $\Omega \subset \mathbb{R}^N$ is a bounded smooth domain, and $\chi_i,a_i,b_i,c_i$ ($i=1,2$) and $\mu,\nu,\lambda$ are positive constants. Among others, we prove that for any given nonnegative initial data $u_0,v_0\in C^0(\bar\Omega)$ with $u_0+v_0\not \equiv 0$, (0.1) has a unique globally defined classical solution $(u(t,x;u_0,v_0),v(t,x;u_0,v_0),w(t,x;u_0,v_0))$ with $u(0,x;u_0,v_0)=u_0(x)$ and $v(0,x;u_0,v_0)=v_0(x)$ in any space dimensional setting. Moreover, we prove that there is $\chi^*(\mu,\chi_1,\chi_2)>0$ satisfying \begin{align*}
&\chi^*(\mu,\chi_1,\chi_2)= \begin{cases}
\frac{\mu \chi^2}{4} &{\rm if}\,\, 0<\chi<2\cr \mu(\chi-1) &{\rm if}\,\, \chi\ge 2, \end{cases} \quad \quad {\rm when \, \chi_1=\chi_2 :=\chi}\\ \text{and}\\ &\chi^*(\mu,\chi_1,\chi_2) \leq \min\Big\{\mu \chi_2+\frac{\mu (\chi_1-\chi_2)^2}{4\chi_2},\mu\chi_1+\frac{\mu(\chi_2-\chi_1)^2}{4\chi_1}\Big\}, \quad \quad {\rm when \, \chi_1 \not=\chi_2} \end{align*} such that the condition $$ \min\{a_1,a_2\}>\chi^*(\mu,\chi_1,\chi_2) $$ implies $$\limsup_{t\rightarrow\infty} \|u(t,\cdot;u_0,v_0)+v(t,\cdot;u_0,v_0)\|_\infty\leq M^*\,{\rm and}\,\liminf_{t\rightarrow\infty}\inf_{x\in\Omega}(u(t,x,u_0,v_0)+v(t,x;u_0,v_0))\ge m^*$$ for some positive constants $M^*,m^*$ independent of $u_0,v_0$, the latter is referred to as combined pointwise persistence.