Boundedness in a three-component quasilinear chemotaxis system on Alopecia Areata

In this paper, we consider a three component quasilinear chemotaxis system for alopecia areata$ \begin{equation*} \begin{cases} u_t = \nabla\cdot(D_1(u)\nabla u)-\chi_1\nabla\cdot(S_1(u)\nabla w)+w-\mu_1u^{\gamma_1}, &x\in\Omega,t>0,\\ v_t = \nabla\cdot(D_2(v)\nabla v)-\chi_2\nabla\cdot(S_2(v)\nabla w)+w+ruv-\mu_2v^{\gamma_2},&x\in\Omega,t>0,\\ w_t = \Delta w+u+v-w, &x\in\Omega,t>0, \end{cases} \end{equation*} $in a smoothly bounded domain $ \Omega\subset\mathbb{R}^n(n\geq1) $ with Neumman boundary conditions, where parameters $ \chi_i,\mu_i\; (i = 1,2) $ and $ r $ are positive. The functions $ D_i(\cdot) $ and $ S_i(\cdot) $ belong to $ C^2 $ satisfying $ D_i(s)\ge(s+1)^{\alpha_i} $ and $ S_i(s)\le s(s+1)^{\beta_i-1} $ with $ \alpha_i,\beta_i\in \mathbb{ R} $ for all $ s\ge0 $ and $ i = 1,2 $. We study the global boundedness of classical solutions existing without any further restrictions on the size of system parameters in two cases: (i) both the diffusion and the logistic damping balance the cross-diffusion; (ii) the logistic damping inhibits the cross-diffusion. Those results not only extend the existing results by Xu (JMAA, 2023), but also draw some new conclusions.