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:$ {align} \{{aligned} & uₜ = Δ u-χ₁∇·(u∇ w)+w-μ₁u^{r₁}, &&x∈Ω, t>0, \\ & vₜ = Δ v-χ₂∇·(v∇ w)+w+ruv-μ₂v^{r₂}, &&x∈Ω, t>0, \\ & wₜ = Δ w+u+v-w, &&x∈Ω, t>0, \ {aligned}. {align} $which was initially proposed by Dobreva et al. [3] to describe the dynamics of hair loss in Alopecia Areata form. Here, $ Ω⊂ RN $ $ (N≥3) $ is a smooth bounded domain, and the parameters fulfill $ χᵢ>0 $, $ μᵢ>0 $, $ rᵢ≥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 $ r₁ = r₂ = 2 $ and $ min\{μ₁, μ₂\}>μ^{} $ or $ rᵢ>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^{} = {2(N-2)₊}{N}CN/2+1^{{1}{N/2+1}}max\{χ₁, χ₂\}+[(2/N)2/N+2N/N+2]r $, where $ CN/2+1 $ is a positive constant corresponding to the maximal Sobolev regularity. Furthermore, we demonstrate that the basic assumption $ μᵢ>0 $ $ (i = 1, 2) $ is sufficient to guarantee the global existence of weak solutions for $ N≥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.