Global boundedness and asymptotic behavior of the chemotaxis system for Alopecia Areata with weakly singular sensitivity

TL;DR

This study analyzes a chemotaxis model for Alopecia Areata, proving global boundedness and exponential convergence of solutions, supported by numerical experiments.

Contribution

It establishes global boundedness and exponential convergence of solutions for the Alopecia Areata chemotaxis model with weakly singular sensitivity in two dimensions.

Findings

Solutions are globally bounded in two dimensions.

Solutions converge exponentially to a constant steady state.

Numerical experiments support theoretical results.

Abstract

This paper considers the homogeneous Neumann initial-boundary value problem for Alopecia Areata chemotaxis model with weakly singular sensitivity. For any appropriately regular initial conditions,it is shown that the problem admits a global boundedness of classical solutions in two spatial dimensions. Moreover, through the explicit construction of Lyapunov functions, we establish that the globally bounded solution converges exponentially to a constant steady state. The paper concludes with numerical experiments that serve to visually illustrate and corroborate some of the theoretically derived findings.