Some progress in global existence of solutions to a higher-dimensional chemotaxis system modelling Alopecia Areata
Haotian Tang, Jiashan Zheng
TL;DR
The work analyzes a three-component parabolic chemotaxis system modelling Alopecia Areata, described by $u_t=\Delta u-\chi_1\nabla\cdot(u\nabla w)+w-\mu_1 u^{r_1}$, $v_t=\Delta v-\chi_2\nabla\cdot(v\nabla w)+w+ruv-\mu_2 v^{r_2}$, and $w_t=\Delta w+u+v-w$ on a bounded domain with $N\ge 3$ and Neumann boundary conditions. It establishes (i) global existence and boundedness of classical solutions for $N\ge 3$ under either $r_1=r_2=2$ with $\min\{\mu_1,\mu_2\}>\mu^*$, or $r_i>2$ with no restriction on $\mu_i$, where $\mu^* = \frac{2(N-2)_+}{N}C_{\frac{N}{2}+1}^{\frac{1}{\frac{N}{2}+1}}\max\{\chi_1,\chi_2\} + \left[(\frac{2}{N})^{\frac{2}{N+2}}\frac{N}{N+2}\right] r$; and (ii) global existence of weak solutions for any $\mu_i>0$, without convexity assumptions on $\Omega$. The analysis uses energy estimates, maximal Sobolev regularity, $L^p$-bootstrap, and Moser iteration for the classical solution, and employs a regularized problem with $F_{\varepsilon}$ plus Aubin–Lions compactness to obtain weak solutions. These results quantify how generalized logistic damping can prevent blow-up in higher dimensions and advance the weak-solution theory for this class of three-component KS-type systems, with implications for biological modeling of AA.
Abstract
This paper is concerned with different logistic damping effects on the global existence in a chemotaxis system \begin{equation*} \left\{\aligned & u_{t}=Δu-χ_{1}\nabla\cdot(u\nabla w)+w-μ_{1}u^{r_{1}},&&x\inΩ,t>0, & v_{t}=Δv-χ_{2}\nabla\cdot(v\nabla w)+w+ruv-μ_{2}v^{r_{2}},&&x\inΩ,t>0, & w_{t}=Δw+u+v-w,&&x\inΩ,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, $Ω\subset\mathbb R^{N}$ $(N\geq3)$ is a bounded domain with smooth boundary, and the parameters fulfill $χ_{i}>0$, $μ_{i}>0$, $r_{i}\geq2$ $(i=1,2)$ and $r>0$. It is proved that if $r_{1}=r_{2}=2$ and $\min\{μ_{1},μ_{1}\}>μ^{\star}$ or $r_{i}>2$ $(i=1,2)$, the Neumann type initial-boundary value problem admits a unique classical solution which is globally bounded in $Ω\times(0,\infty)$ for all sufficiently smooth initial data. The lower bound $μ^{\ast}=\frac{2(N-2)_{+}}{N}C_{\frac{N}{2}+1}^{\frac{1}{\frac{N}{2}+1}}\max\{χ_{1},χ_{2}\}+\left[(\frac{2}{N})^{\frac{2}{N+2}}\frac{N}{N+2}\right]r$, where $C_{\frac{N}{2}+1}$ is a positive constant corresponding to the maximal Sobolev regularity. Furthermore, the basic assumption $μ_{i}>0$ $(i=1,2)$ 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.