Boundedness and asymptotic stability of solutions in an alopecia areata chemotaxis system with signal-dependent sensitivity

This work deals with a three-component chemotaxis system \(\begin{aligned} \left\{ \begin{array}{llll} u_t=\Delta u-\nabla \cdot (u\chi _1(w)\nabla w)+w-\mu _1u^2,\quad & x\in \Omega ,\quad t>0,\\ v_t=\Delta v-\nabla \cdot (v\chi _2(w)\nabla w)+w+ruv-\mu _2v^2,\quad & x\in \Omega ,\quad t>0,\\ w_t=\Delta w+u+v-w,\quad & x\in \Omega ,\quad t>0,\\ \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{u}_t=\mathrm{\Delta }u-\mathrm{\nabla }·(u{\chi }_1(w)\mathrm{\nabla }w)+w-{\mu }_1{u}^2,\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ v_t=\mathrm{\Delta }v-\mathrm{\nabla }·(v{\chi }_2(w)\mathrm{\nabla }w)+w+ruv-{\mu }_2{v}^2,\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ w_t=\mathrm{\Delta }w+u+v-w,\hfill & x\in \mathrm{\Omega },t>0,\hfill \end{array}\right)\end{array}$$ describing the spatiotemporal Alopecia Areata dynamics with signal-dependent sensitivity under homogeneous Neumann boundary conditions in a smoothly bounded domain \(\Omega \subset \mathbb {R}^n(n\ge 1)\) $\mathrm{\Omega }\subset {\mathbb{R}}^n(n\ge 1)$ , where the parameters \(\mu _1\) ${\mu }_1$ , \(\mu _2\) ${\mu }_2$ , r are positive constants. The chemotactic function \(\chi _i(w)(i=1, 2)\) ${\chi }_i(w)(i=1,2)$ is smooth and satisfying some conditions. Given suitably regular initial data, it is shown that the corresponding initial-boundary value problem admits a unique global bounded classical solution. Furthermore, by constructing the appropriate energy function, it is proved that the unique coexistence equilibrium is globally convergent in a specific parameter regime. These results generalize and improve some well-known results in the literature.