Boundedness and asymptotic behavior in a quasilinear chemotaxis system for alopecia areata

In this paper, we study the following chemotaxis system with generalized volume-filling effect ut=∇⋅(D1(u)∇u)−χ1∇⋅(S1(u)∇w)+w−μ1uη1,(x,t)∈Ω×(0,∞),vt=∇⋅(D2(v)∇v)−χ2∇⋅(S2(v)∇w)+w+ruv−μ2vη2,(x,t)∈Ω×(0,∞),0=Δw+u+v−w,(x,t)∈Ω×(0,∞),under homogeneous Neumann boundary conditions in a smoothly bounded domain Ω⊂Rn (n≥1), which describes the spatio-temporal dynamics of alopecia areata lesions, where χ1, χ2, r, μ1, μ2 are positive parameters and η1,η2≥2. When the functions Di and Si (i=1,2) belong to C2 fulfilling Di(s)≥(s+1)αi, 0≤S(s)≤sβi with αi∈R and βi≥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 ηi=2 and 0≤Si(s)≤s(s+1)βi−1 with 0≤βi<1 or 0≤Si(s)≤sβi with βi≥1 for all s≥0 and i=1,2, if the chemosensitivity χ1 and χ2 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.