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

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 and autocrine loops:\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, &\qquad x\in\Omega, \, t>0, \\ &w_t = \nabla\cdot(D_2(w)\nabla w-S_2(w)\nabla z-S_3(w)\nabla v), &\qquad x\in\Omega, \, t>0, \\ & \tau_2 z_t = \Delta z- z+ u, &\qquad x\in\Omega, \, t>0, \end{aligned} \right. \end{equation*} $\end{document}where \begin{document}$ u(t, x) $\end{document} and \begin{document}$ w(t, x) $\end{document} denote the density of macrophages and tumor cells at time \begin{document}$ t $\end{document} and location \begin{document}$ x\in \Omega, $\end{document} respectively, \begin{document}$ v(t, x) $\end{document} and \begin{document}$ z(t, x) $\end{document} represent the concentration of colony stimulating factor 1 (CSF-1) secreted by the tumor cells and epidermal growth factor (EGF) secreted by macrophages at time \begin{document}$ t $\end{document} and location \begin{document}$ x\in \Omega, $\end{document} respectively. \begin{document}$ \Omega\subset \mathbb{R}^n $\end{document} is a bounded region with smooth boundary, \begin{document}$ \tau_i\ge 0 \; (i = 1, 2) $\end{document} , \begin{document}$ D_i(s)\ge d_i(s+1)^{m_i-1} $\end{document} with parameters \begin{document}$ m_i\ge 1 \; (i = 1, 2) $\end{document} and \begin{document}$ S_j(s)\lesssim (s+1)^{q_j} $\end{document} with parameters \begin{document}$ q_j>0 \;(j = 1, 2, 3) $\end{document} . For the case without autocrine loop (i.e., \begin{document}$ S_3(w) = 0 $\end{document} ), it is shown that when \begin{document}$ q_j\le 1 \; (j = 1, 2) $\end{document} , if one of \begin{document}$ q_j $\end{document} is smaller than one or one of \begin{document}$ m_i $\end{document} is larger than one, then the IBVP has a global classical solution which is uniformly bounded. Moreover, when \begin{document}$ m_1 = m_2 = q_1 = q_2 = 1 $\end{document} , an inequality involving the product \begin{document}$ d_1d_2 $\end{document} and the product of the two species' initial mass is obtained which guarantees the existence of global bounded classical solutions. More specifically, it allows one of \begin{document}$ d_i $\end{document} to be small or one of the species initial mass to be large. For the case with autocrine loop (i.e \begin{document}$ S_3(w)\ne 0 $\end{document} ), similar results hold only if \begin{document}$ q_3<1 $\end{document} . If \begin{document}$ q_3 = 1 $\end{document} , solutions to the IBVP exist globally only when \begin{document}$ d_2 $\end{document} is suitably large or the mass of species \begin{document}$ w $\end{document} is suitably small.