Boundedness in the higher-dimensional chemotaxis system for Alopecia Areata with singular sensitivity

This paper deals with a fully parabolic chemotaxis system $\left(\begin{array}{cc}{u}_t=\Delta u-{\chi }_1\nabla \cdot (\frac{u}{w}\nabla w)+w-{\mu }_1{u}^2,& x\in \Omega ,t>0,\\ v_t=\Delta v-{\chi }_2\nabla \cdot (\frac{v}{w}\nabla w)+w+ruv-{\mu }_2{v}^2,& x\in \Omega ,t>0,\\ w_t=\Delta w+u+v-w,& x\in \Omega ,t>0\end{array}\right)$in a bounded domain $\Omega \subset R^n(n\ge 2)$ with smooth boundary, where $r>0,{\chi }_i>0,{\mu }_i>0(i=1,2).$ Under the condition ${\mu }_1\ge 2r,{\mu }_2\ge 2r,max\{{\chi }_1,{\chi }_2\}<\sqrt{\frac{2}{n}},$ it is shown that the corresponding system possesses a globally bounded classical solution, which extends the previous boundedness result from $n=2$ to $n\ge 2$.