Some progress in global existence of solutions to a higher-dimensional chemotaxis system modelling Alopecia Areata

This paper is concerned with different logistic damping effects on the global existence in a chemotaxis system \begin{equation*} \left\{\aligned & u_{t}=\Delta u-\chi_{1}\nabla\cdot(u\nabla w)+w-\mu_{1}u^{r_{1}},&&x\in\Omega,t>0, & v_{t}=\Delta v-\chi_{2}\nabla\cdot(v\nabla w)+w+ruv-\mu_{2}v^{r_{2}},&&x\in\Omega,t>0, & w_{t}=\Delta w+u+v-w,&&x\in\Omega,t>0,\\ \endaligned\right. \end{equation*} which was initially proposed by Dobreva \emph{et al.} (\cite{DP2020}) to describe the dynamics of hair loss in Alopecia Areata form. Here, $\Omega\subset\mathbb R^{N}$ $(N\geq3)$ is a bounded domain with smooth boundary, and the parameters fulfill $\chi_{i}>0$, $\mu_{i}>0$, $r_{i}\geq2$ $(i=1,2)$ and $r>0$. It is proved that if $r_{1}=r_{2}=2$ and $\min\{\mu_{1},\mu_{1}\}>\mu^{\star}$ or $r_{i}>2$ $(i=1,2)$, the Neumann type initial-boundary value problem admits a unique classical solution which is globally bounded in $\Omega\times(0,\infty)$ for all sufficiently smooth initial data. The lower bound $\mu^{\ast}=\frac{2(N-2)_{+}}{N}C_{\frac{N}{2}+1}^{\frac{1}{\frac{N}{2}+1}}\max\{\chi_{1},\chi_{2}\}+\left[(\frac{2}{N})^{\frac{2}{N+2}}\frac{N}{N+2}\right]r$, where $C_{\frac{N}{2}+1}$ is a positive constant corresponding to the maximal Sobolev regularity. Furthermore, the basic assumption $\mu_{i}>0$ $(i=1,2)$ can ensure the global existence of a weak solution. Notably, our findings not only first provide new insights into the weak solution theory of this system but also offer some novel quantized impact of the (generalized) logistic source on preventing blow-ups.