BOUNDEDNESS IN A THREE-COMPONENT QUASILINEAR CHEMOTAXIS SYSTEM ON ALOPECIA AREATA

In this paper, we consider a three component quasilinear chemo-taxis system for alopecia areata { ut = V <middle dot> (D1(u)Vu) - chi 1V <middle dot> (S1(u)Vw) + w - mu 1u gamma 1 , x E Omega, t > 0,vt = V <middle dot> (D2(v)Vv) - chi 2V <middle dot> (S2(v)Vw) + w + ruv - mu 2v gamma 2, x E Omega, t > 0,wt= Delta w + u + v- w, x E Omega, t > 0,in a smoothly bounded domain Omega c R-n(n > 1) with Neumman boundary conditions, where parameters chi i, mu i (i = 1,2) and r are positive. The functions Di(<middle dot>) and Si(<middle dot>) belong to C-2 satisfying Di(s) > (s + 1)(alpha i) and Si(s) < s(s + 1)(beta i-1) with alpha(i), beta(i)is an element of R for all s > 0 and i = 1, 2. We study the global boundedness of classical solutions existing without any further restrictions on the size of system parameters in two cases: (i) both the diffusion and the logistic damping balance the cross-diffusion; (ii) the logistic damping inhibits the cross-diffusion. Those results not only extend the existing results by Xu (JMAA, 2023), but also draw some new conclusions.