Summary: This paper deals with a two-competing-species chemotaxis system with indirect signal consumption \[ \begin{cases} \begin{aligned} &u_t=d_1\Delta u-\chi_1\nabla \cdot (u\nabla w)+\mu_1u(1-u-a_1v),&(x,t)\in \Omega \times (0,\infty ),\\ &v_t=d_2\Delta v-\chi_2\nabla \cdot (v\nabla w)+\mu_2v(1-v-a_2u),&(x,t)\in \Omega \times (0,\infty ),\\ &w_t=\Delta w-wz,&(x,t)\in \Omega \times (0,\infty ),\\ &z_t=\Delta z-z+u+v,&(x,t)\in \Omega \times (0,\infty ),\\ &\frac{{\partial u}}{{\partial \nu }} = \frac{{\partial v}}{{\partial \nu }} = \frac{{\partial w}}{{\partial \nu }} = \frac{{\partial z}}{{\partial \nu }} = 0,&(x,t)\in \partial \Omega \times (0,\infty ),\\ &\left( {u, v, w, z} \right) \left( {x,0} \right) = \left( {{u_0}\left( x \right) , {v_0}\left( x \right) ,{w_0}\left( x \right) , {z_0}\left( x \right) } \right) ,&x\in \Omega ,\ \end{aligned} \end{cases}\] under homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega \subset{\mathbb{R}}^n(n\le 2)\), with the nonnegative initial data \(\left( {u_0, v_0, w_0, z_0} \right) \in{C^0}\left( {{{\bar{\Omega }}} } \right) \times{C^0}\left( {{{\bar{\Omega }}} } \right) \times{W^{1,\infty }}\left( \Omega \right) \times{W^{1,\infty }}\left( \Omega \right) \), where \(\chi_i>0, d_i>0, a_i>0\) and \(\mu_i>0 (i=1,2)\). It is shown that the system has a global bounded classical solution for arbitrary size of \(\mu_1, \mu_2>0\). Additionally, we consider the asymptotic stabilization of solutions to the above system as follows:

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When \(a_1, a_2 \in (0,1)\), the global bounded classical solution \((u, v, w, z)\) exponentially converges to \(\Big (\frac{1-a_1}{1-a_1 a_2}, \frac{1-a_2}{1-a_1 a_2}, 0, \frac{2-a_1-a_2}{1-a_1 a_2}\Big )\) in the \(L^{\infty } \)-norm as \(t \rightarrow \infty \);

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When \(a_1>1>a_2>0\) and \(a_1a_2<1\), the global bounded classical solution \((u, v, w, z)\) exponentially converges to (0, 1, 0, 1) in the \(L^{\infty } \)-norm as \(t \rightarrow \infty \);

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When \(a_1=1>a_2>0\), the global bounded classical solution \((u, v, w, z)\) polynomially converges to (0, 1, 0, 1) in the \(L^{\infty } \)-norm as \(t \rightarrow \infty \).