Global existence, boundedness and asymptotic behavior of classical solutions of a two-species chemotaxis-competition system with two chemicals on <inline-formula><tex-math id="M1">$ \mathbb{R}^N $</tex-math></inline-formula>

In this paper, we investigate the following two-species chemotaxis-competition system with two chemicals on $ \mathbb{R}^N $ $ \begin{align} \begin{cases} u_t = \Delta u-\chi_1\nabla\cdot(u\nabla v)+u(a_1-b_1u-c_1w), &t>0, \; x\in\mathbb{R}^N, \\ 0 = \Delta v- v+ w, &t>0, \; x\in\mathbb{R}^N, \\ w_t = \Delta w-\chi_2\nabla\cdot(w\nabla z)+w(a_2-b_2w-c_2u), &t>0, \; x\in\mathbb{R}^N, \\ 0 = \Delta z-z+u, &t>0, \; x\in\mathbb{R}^N, \\ u(0, x) = u_0(x), \; w(0, x) = w_0(x), &x\in\mathbb{R}^N, \end{cases} \end{align} ~~~~(1)$ where $ N\geq1 $ is a positive integer, $ \chi_i, a_i, b_i, c_i (i = 1, 2) $ are positive constants and the initial data $ u_0, w_0\in C^b_{\text{unif}}(\mathbb{R}^N) $ or $ u_0, w_0\in L^q(\mathbb{R}^N) $ with $ q>n $ and $ q\geq2 $, and $ u_0\geq0, $ $ w_0\geq0. $ Suppose that $ (u, v, w, z) $ is a nonnegative classical solution of (1) on the maximal interval $ (0, T_{\max}). $ We first prove that if there exists $ p>\max\{1, \frac{N}{2}\} $ such that $ \begin{align} \begin{split} \limsup\limits_{t\rightarrow T_{\max}-}\sup\limits_{x_0\in\mathbb{R}^N}\int_{B(x_0, 1)}u^p(t, x)dx<\infty, \\ \limsup\limits_{t\rightarrow T_{\max}-}\sup\limits_{x_0\in\mathbb{R}^N}\int_{B(x_0, 1)}w^p(t, x)dx<\infty, \end{split} \end{align} $ then $ T_{\max} = \infty $ and $ \limsup_{t\rightarrow \infty}(\|u(t, \cdot)\|_{\infty}+\|w(t, \cdot)\|_{\infty})<\infty. $ Moreover, under the sufficient conditions$ \chi_1<c_1\cdot\frac{N}{(N-2)_+}, \; \; \; \chi_2<c_2\cdot\frac{N}{(N-2)_+}, $we prove the nonnegative classical solution of (1) exists globally and stays bounded. Finally, we discuss the asymptotic behavior of classical solution of (1) in three competition cases.