On a macrophage and tumor cell chemotaxis system with both paracrine and autocrine loops

<p style='text-indent:20px;'>In this paper, we consider a homogeneous Neumann initial-boundary value problem (IBVP) for the following two-species and two-stimuli chemotaxis model with both paracrine autocrine loops:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \label{IBVP} \left\{ \begin{aligned} &u_t = \nabla\cdot(D_1(u)\nabla u-S_1(u)\nabla v), &\qquad x\in\Omega, \, t>0, \\ & \tau_1 v_t \Delta v- v+w, &w_t \nabla\cdot(D_2(w)\nabla w-S_2(w)\nabla z-S_3(w)\nabla \tau_2 z_t z- z+ u, \end{aligned} \right. \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ u(t, x) $\end{document}</tex-math></inline-formula> id="M2">\begin{document}$ w(t, denote density of macrophages tumor cells at time id="M3">\begin{document}$ t location id="M4">\begin{document}$ x\in \Omega, respectively, id="M5">\begin{document}$ v(t, id="M6">\begin{document}$ z(t, represent concentration colony stimulating factor 1 (CSF-1) secreted by epidermal growth (EGF) id="M7">\begin{document}$ id="M8">\begin{document}$ respectively. id="M9">\begin{document}$ \Omega\subset \mathbb{R}^n is bounded region smooth boundary, id="M10">\begin{document}$ \tau_i\ge 0 \; (i 1, 2) $\end{document}</tex-math></inline-formula>, id="M11">\begin{document}$ D_i(s)\ge d_i(s+1)^{m_i-1} parameters id="M12">\begin{document}$ m_i\ge id="M13">\begin{document}$ S_j(s)\lesssim (s+1)^{q_j} id="M14">\begin{document}$ q_j>0 \;(j 2, 3) $\end{document}</tex-math></inline-formula>. For case without loop (i.e., id="M15">\begin{document}$ S_3(w) $\end{document}</tex-math></inline-formula>), it shown that when id="M16">\begin{document}$ q_j\le (j if one id="M17">\begin{document}$ q_j smaller than or id="M18">\begin{document}$ m_i larger one, then IBVP has global classical solution which uniformly bounded. Moreover, id="M19">\begin{document}$ m_1 m_2 q_1 q_2 an inequality involving product id="M20">\begin{document}$ d_1d_2 two species' initial mass obtained guarantees existence solutions. More specifically, allows id="M21">\begin{document}$ d_i to be small species large. (i.e id="M22">\begin{document}$ S_3(w)\ne similar results hold only id="M23">\begin{document}$ q_3<1 If id="M24">\begin{document}$ q_3 solutions exist globally id="M25">\begin{document}$ d_2 suitably large id="M26">\begin{document}$ w small.</p>