Global solvability in a higher-dimensional chemotaxis system for alopecia areata: Nonlinear proliferation versus logistic degradation

Abstract

This paper is concerned with the Neumann initial-boundary value problem for the chemotaxis system:$ \begin{align} \left\{\begin{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, \ \end{aligned}\right. \end{align} $which was initially proposed by Dobreva et al. [3] to describe the dynamics of hair loss in Alopecia Areata form. Here, $ \Omega\subset\mathbb R^{N} $ $ (N\geq3) $ is a smooth bounded domain, and the parameters fulfill $ \chi_{i}>0 $, $ \mu_{i}>0 $, $ r_{i}\geq2 $ $ (i = 1, 2) $ and $ r>0 $. The inherent presence of two positive chemotaxis terms, along with the zero-order nonlinear production term $ ruv $, significantly complicates the energy estimation. It is proved that if $ r_{1} = r_{2} = 2 $ and $ \min\{\mu_{1}, \mu_{2}\}>\mu^{\star} $ or $ r_{i}>2 $ $ (i = 1, 2) $, this problem admits a global bounded classical solution for all sufficiently smooth initial data. The lower bound is given by $ \mu^{\star} = \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, we demonstrate that the basic assumption $ \mu_{i}>0 $ $ (i = 1, 2) $ is sufficient to guarantee the global existence of weak solutions for $ N\geq3 $. Notably, our findings not only extend or refine several existing results (see Remarks 1.1-1.2) but also provide new insights into the weak solution theory of this system for the first time.