Global solvability in a higher-dimensional chemotaxis system for Alopecia Areata: Nonlinear proliferation versus logistic degradation
This paper is concerned with the Neumann initial-boundary value problem for the chemotaxis system: $u_t=Δu-χ_1\nabla\cdot(u\nabla w)+w-μ_1u^{r_1}$, $v_t=Δv-χ_2\nabla\cdot(v\nabla w)+w+ruv-μ_2v^{r_2}$, and $w_t=Δw+u+v-w$ in $Ω\times(0,\infty)$, which was initially proposed by Dobreva et al. to describe the dynamics of hair loss in Alopecia Areata form. Here, $Ω\subset\mathbb R^{N}$ $(N\geq3)$ is a smooth bounded domain, and the parameters fulfill $χ_{i}>0$, $μ_{i}>0$, $r_{i}\geq2$ $(i=1,2)$ 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 $\min\{μ_{1},μ_{2}\}>μ^{\star}$ 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 $μ^{\star}=\frac{2(N-2)_{+}}{N}C_{\frac{N}{2}+1}^{\frac{1}{\frac{N}{2}+1}}\max\{χ_{1},χ_{2}\}+\left[(\frac{2}{N})^{\frac{2}{N+2}}\frac{N}{N+2}\right]r$, where $C_{\frac{N}{2}+1}$ is a positive constant corresponding to the maximal Sobolev regularity. Furthermore, we demonstrate that the basic assumption $μ_{i}>0$ $(i=1,2)$ 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.