Boundedness and asymptotic behavior in a quasilinear chemotaxis system with nonlinear diffusion and singular sensitivity for alopecia areata

This paper is concerned with a three-component quasilinear chemotaxis system with nonlinear diffusion and singular sensitivity for alopecia areata ut=∇D1(u)∇u−χ1∇uw∇w+w−μ1uγ1,(x,t)∈Ω×(0,∞),vt=∇D2(v)∇v−χ2∇vw∇w+w+ruv−μ2vγ2,(x,t)∈Ω×(0,∞),wt=Δw+u+v−w,(x,t)∈Ω×(0,∞),∂u∂ν=∂v∂ν=∂w∂ν=0,(x,t)∈∂Ω×(0,∞),u(x,0)=u0(x),v(x,0)=v0(x),w(x,0)=w0(x),x∈Ω,associated with homogeneous Neumann boundary conditions in a convex smooth bounded domain Ω⊂R2. For i=1,2, the parameters χi,μi,r are positive and γi≥2. The nonlinear diffusion functions Di(s)∈C2 satisfy Di(s)⩾(s+1)αi for all s≥0. We delve into analyzing the global existence and boundedness of classical solutions for the aforementioned system under specific conditions. Additionally, in the scenario where γi=2, we develop a Lyapunov functional and scrutinize its temporal evolution to ascertain the asymptotic stability of the coexistence state.