Global boundedness and asymptotic behavior of the chemotaxis system for alopecia areata with singular sensitivity

Abstract

This paper is concerned with a three-component chemotaxis system for alopecia areata with singular sensitivity ut=Δu−χ1∇⋅uw∇w+w−μ1u2,x∈Ω,t>0,vt=Δv−χ2∇⋅vw∇w+w+ruv−μ2v2,x∈Ω,t>0,wt=Δw+u+v−w,x∈Ω,t>0,∂u∂ν=∂v∂ν=∂w∂ν=0,x∈∂Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),w(x,0)=w0(x),x∈Ωunder the homogeneous Neumann boundary conditions in a smoothly bounded domain Ω⊂R2, where the parameters χi, μi(i=1,2) and r are positive. It is showed that if χ1,χ2<52, this system admits a globally bounded classical solution. Furthermore, under the particular conditions of μ1<μ2<3μ1 and r=μ2−μ1, the global bounded solution converges to the steady state (2μ1,2μ1,4μ1) as t→∞.