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

2025/06/04 by Haotian Tang, H.Y. Tang, Tang, Haotian +2
Engineering · Mathematics · Medicine · #Mathematical Biology Tumor Growth #Mathematical and Theoretical Epidemiology and Ecology Models #Thermoelastic and Magnetoelastic Phenomena #math.AP

paper · pdf · doi:10.48550/arxiv.2506.03565

openalex publication_date 2025/06/04 · openalex created_date 2025/10/10 · arxiv created 2026/07/29 · arxiv updated 2026/07/30 · openalex updated_date 2026/08/02

Abstract

This paper is concerned with the Neumann initial-boundary value problem for the chemotaxis system: u_t=Δu-χ₁∇⋅(u∇ w)+w-μ₁u^r₁, v_t=Δv-χ₂∇⋅(v∇ w)+w+ruv-μ₂v^r₂, and w_t=Δw+u+v-w in Ω×(0,∞), which was initially proposed by Dobreva et al. to describe the dynamics of hair loss in Alopecia Areata form. Here, Ω⊂\mathbb R^N (N≥3) is a smooth bounded domain, and the parameters fulfill χ_i>0, μ_i>0, r_i≥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 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^⋆=\frac2(N-2)₊NC_(N)/(2)+1^{(1)/((N)/(2)+1)}max\χ₁,χ₂\+[((2)/(N))^(2)/(N+2)(N)/(N+2)]r, where C_(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≥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.

Citations

• Stability and instability in a three-component chemotaxis model for alopecia areata
• Chemotactic Aggregation versus Logistic Damping on Boundedness in the 3D Minimal Keller--Segel Model
• A new result for global existence and boundedness of solutions to a parabolic--parabolic Keller--Segel system with logistic source
• A note for global existence of a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant
• Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source

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