Global boundedness of a higher-dimensional chemotaxis system on alopecia areata

Abstract

This paper mainly focuses on the dynamics behavior of a three-component chemotaxis system on alopecia areata $ \begin{equation*} \left\{ \begin{array}{lll} u_t = \Delta{u}-\chi_1\nabla\cdot(u\nabla{w})+w-\mu_1u^2, &x\in\Omega, t>0, \\ v_t = \Delta{v}-\chi_2\nabla\cdot(v\nabla{w})+w+ruv-\mu_2v^2, &x\in \Omega, t>0, \\ w_t = \Delta{w}+u+v-w, &x\in \Omega, t>0, \\ \frac{\partial{u}}{\partial{\nu}} = \frac{\partial{v}}{\partial{\nu}} = \frac{\partial{w}}{\partial{\nu}} = 0, &x\in \partial \Omega, t>0, \\ u(x, 0) = u_0(x), \ v(x, 0) = v_0(x), \ w(x, 0) = w_0(x), &x\in \Omega, \ \end{array} \right. \end{equation*} $ where $ \Omega\subset\mathbb{R}^n $ $ (n \geq 4) $ is a bounded convex domain with smooth boundary $ \partial\Omega $, the parameters $ \chi_i $, $ \mu_i $ $ (i = 1, 2) $, and $ r $ are positive. We show that this system exists a globally bounded classical solution if $ \mu_i\; (i = 1, 2) $ is large enough. This result extends the corresponding results which were obtained by Lou and Tao (JDE, 2021) to the higher-dimensional case.

Keywords

• chemotaxis
• alopecia areata
• global existence