Boundedness and asymptotic behavior in a quasilinear chemotaxis system for alopecia areata. (August 2023) - Shan, Wenhai; Zheng, Pan
Abstract: In this paper, we study the following chemotaxis system with generalized volume-filling effect u t = ∇ ⋅ ( D 1 ( u ) ∇ u ) − χ 1 ∇ ⋅ ( S 1 ( u ) ∇ w ) + w − μ 1 u η 1, ( x, t ) ∈ Ω × ( 0, ∞ ), v t = ∇ ⋅ ( D 2 ( v ) ∇ v ) − χ 2 ∇ ⋅ ( S 2 ( v ) ∇ w ) + w + r u v − μ 2 v η 2, ( x, t ) ∈ Ω × ( 0, ∞ ), 0 = Δ w + u + v − w, ( x, t ) ∈ Ω × ( 0, ∞ ), under homogeneous Neumann boundary conditions in a smoothly bounded domain Ω ⊂ R n ( n ≥ 1 ), which describes the spatio-temporal dynamics of alopecia areata lesions, where χ 1, χ 2, r, μ 1, μ 2 are positive parameters and η 1, η 2 ≥ 2 . When the functions D i and S i ( i = 1, 2 ) belong to C 2 fulfilling D i ( s ) ≥ ( s + 1 ) α i, 0 ≤ S ( s ) ≤ s β i with α i ∈ R and β i ≥ 0 for all s ≥ 0, we study the global existence and boundedness of classical solutions for the above system under some suitable conditions, and find that either the higher-order nonlinear diffusion or strong logistic damping can prevent blow-up of classical solutions for the problem. In addition, when η i = 2 and 0 ≤ S i ( s ) ≤ s ( s + 1 ) β i − 1 with 0 ≤ β i < 1 or 0 ≤ S i ( s ) ≤ s β i with β i ≥ 1 for all s ≥ 0 and i = 1, 2, if the chemosensitivity χ 1 and χ 2 are appropriately mild and some other parameter conditions hold, the globally bounded solution will stabilize to the constant coexistence equilibrium as the time goes to infinity. Our results not only extend the previous ones of Tao & Xu (2022), but also involve some new conclusions.