Global boundedness and asymptotic behavior of the chemotaxis system for alopecia areata with singular sensitivity

This paper is concerned with a three-component chemotaxis system for alopecia areata with singular sensitivity $\left(\begin{array}{cc}{u}_t=\Delta u-{\chi }_1\nabla \cdot \left(\frac{u}{w}\nabla w\right)+w-{\mu }_1{u}^2,& x\in \Omega ,t>0,\\ v_t=\Delta v-{\chi }_2\nabla \cdot \left(\frac{v}{w}\nabla w\right)+w+ruv-{\mu }_2{v}^2,& x\in \Omega ,t>0,\\ w_t=\Delta w+u+v-w,& x\in \Omega ,t>0,\\ \frac{\partial u}{\partial \nu }=\frac{\partial v}{\partial \nu }=\frac{\partial w}{\partial \nu }=0,& x\in \partial \Omega ,t>0,\\ u(x,0)=u_0\left(x\right),v(x,0)=v_0\left(x\right),w(x,0)=w_0\left(x\right),& x\in \Omega \end{array}\right)$under the homogeneous Neumann boundary conditions in a smoothly bounded domain $\Omega \subset R^2$, where the parameters ${\chi }_i$, ${\mu }_i$ $(i=1,2)$ and $r$ are positive. It is showed that if ${\chi }_1,{\chi }_2<\frac{\sqrt{5}}{2}$, this system admits a globally bounded classical solution. Furthermore, under the particular conditions of ${\mu }_1<{\mu }_2<3{\mu }_1$ and $r={\mu }_2-{\mu }_1$, the global bounded solution converges to the steady state $(\frac{2}{{\mu }_1},\frac{2}{{\mu }_1},\frac{4}{{\mu }_1})$ as $t\to \infty$.