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: $u_t=\hat{I}”u-\ddot{I}{‡}_1\nabla \cdot (u\nabla w)+w-\hat{I}{¼}_1{u}^{r_1}$, $v_t=\hat{I}”v-\ddot{I}{‡}_2\nabla \cdot (v\nabla w)+w+ruv-\hat{I}{¼}_2{v}^{r_2}$, and $w_t=\hat{I}”w+u+v-w$ in $\hat{I}\text{c◯}\times (0,\infty )$, which was initially proposed by Dobreva et al. to describe the dynamics of hair loss in Alopecia Areata form. Here, $\hat{I}\text{c◯}\subset {\mathbb{R}}^N$ $(N\ge 3)$ is a smooth bounded domain, and the parameters fulfill $\ddot{I}{‡}_i>0$, $\hat{I}{¼}_i>0$, $r_i\ge 2$ $(i=1,2)$ 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 $\min \{\hat{I}{¼}_1,\hat{I}{¼}_2\}>\hat{I}{¼}^{\star }$ 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 $\hat{I}{¼}^{\star }=\frac{2(N-2{)}_{+}}{N}{C}_{\frac{N}{2}+1}^{\frac{1}{\frac{N}{2}+1}}\max \{\ddot{I}{‡}_1,\ddot{I}{‡}_2\}+\left[(\frac{2}{N}{)}^{\frac{2}{N+2}}\frac{N}{N+2}\right]r$, where $C_{\frac{N}{2}+1}$ is a positive constant corresponding to the maximal Sobolev regularity. Furthermore, we demonstrate that the basic assumption $\hat{I}{¼}_i>0$ $(i=1,2)$ is sufficient to guarantee the global existence of weak solutions for $N\ge 3$. 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.