On a quasilinear two-species chemotaxis system with general kinetic functions and interspecific competition

In this paper, we study the following two-species chemotaxis system with generalized volume-filling effect and general kinetic functions

$$\begin{aligned} \left\{ \begin{aligned} &u_t=\nabla \cdot (D_{1}(u)\nabla u)- \nabla \cdot ( \chi _{1}(u)\nabla w) + f_{1}(u)-\mu _{1}a_{1}uv,&(x,t)\in \Omega \times (0,\infty ), \\&v_t=\nabla \cdot (D_{2}(v)\nabla v)- \nabla \cdot ( \chi _{2}(v)\nabla w) + f_{2}(v)-\mu _{2}a_{2}uv,&(x,t)\in \Omega \times (0,\infty ), \\&\tau w_t=\Delta w - w + g_{1}(u) + g_{2}(v),&(x,t)\in \Omega \times (0,\infty ),\\ \end{aligned} \right. \end{aligned}$$

under homogeneous Neumann boundary conditions in a smoothly bounded domain \(\Omega \subset {\mathbb {R}}^{n}\) \((n\ge 1)\), where \(a_{1}, a_{2}, \mu _{1}, \mu _{2}\) are positive constants. When the functions \(D_{i}, S_{i}, f_{i}, g_{i}\) \((i=1,2)\) belong to \(C^{2}\) fulfilling some suitable hypotheses, we study the global existence and boundedness of classical solutions for the above system and find that under the case of \(\tau =1\) or \(\tau =0\), either the higher-order nonlinear diffusion or strong logistic damping can prevent blow-up of classical solutions for the problem. In addition, when the functions are replaced to Lotka–Volterra competitive kinetic functional response term and linear signal generations, by constructing some appropriate Lyapunov functionals, we show that the solution convergences to the constant steady state in \(L^{\infty }(\Omega )\) in the case of \(a_1, a_2 \in (0,1)\) or \(a_1 \ge 1>a_2 > 0\) under some more concise conditions than [2], which improved the existing conditions to some extent.