Global solvability and stability of a quasilinear chemotaxis system for alopecia areata

This study explores a three-component chemotaxis model for alopecia areata (AA), incorporating nonlinear diffusion and chemosensitivity functions, as well as general kinetic functions. Under some suitable conditions on the above functions and homogeneous Neumann boundary conditions, we first investigate the global existence and boundedness of classical solutions for the AA system, and find that the higher-order nonlinear diffusion of T cells can prevent the classical solutions of AA system from blow-up when the nonlinear diffusion and chemosensitivity meet the volume filling effect. Additionally, the strong damping effect of T cells can ensure the global boundedness of the solution for AA system irrespective of whether the nonlinear diffusion and chemosensitivity meet the volume filling effect or signal-dependence. In addition, by spectral analysis, we discuss the effect of chemotactic strengths on the stability of AA system around the positive constant equilibrium. By numerical simulations, we analyze the potential causes of AA and propose treatment strategies for future studies.