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

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.

• The paper establishes global well-posedness for AA chemotaxis models using energy functional methods to derive uniform bounds.
• It demonstrates exponential convergence to homogeneous steady states with rigorously quantified decay estimates under weakly singular sensitivity (0 < k < 1).
• The work validates theoretical results with numerical simulations that capture Turing instabilities and evolving pattern dynamics in 2D domains.

Global Boundedness and Asymptotic Behavior of a Chemotaxis System with Weakly Singular Sensitivity for Alopecia Areata

Introduction and Mathematical Model

This paper delivers a rigorous mathematical analysis of a chemotaxis system tailored to the pathogenesis of Alopecia Areata (AA), a T-cell driven autoimmune disorder affecting hair follicles. The system under investigation models the densities of CD4$^+$ T cells ($u$), CD8$^+$ T cells ($v$), and interferon-gamma (IFN-$\gamma$, $w$), incorporating weakly singular sensitivity in the chemotactic response with respect to $w$: $$\begin{cases} u_t = \Delta u - \chi_1 \nabla \cdot \left(\frac{u}{w^k} \nabla w \right) + w - \mu_1 u^2, \ v_t = \Delta v - \chi_2 \nabla \cdot \left(\frac{v}{w^k} \nabla w \right) + w + ruv - \mu_2 v^2, \ w_t = \Delta w + u + v - w, \end{cases}$$ on a smooth bounded domain $\Omega \subset \mathbb{R}^2$, with homogeneous Neumann boundary conditions. Here, $\chi_i > 0$ are chemotactic sensitivities, $u$0 encode quadratic damping, $u$1 models cross-species proliferation, and $u$2 quantifies the degree of sensitivity singularity.

The principal theoretical novelty is allowing for weakly singular ($u$3) chemotactic sensitivity, in contrast to the classical strong singular ($u$4) or regular ($u$5) settings. Such weak singularities are motivated by more realistic biochemical signal processing, avoiding technical blow-up challenges while retaining the analytical subtlety of near-zero signal regimes.

Main Analytical Results

Global Existence and Uniform Boundedness

The work establishes that, for arbitrarily large regular initial data, all solution components remain globally well-posed and uniformly bounded for all time. The proof utilizes an energy functional approach: $u$6 allowing the authors to control $u$7 and $u$8 without imposing a strictly positive lower bound on $u$9—a critical obstacle for weak singularities. Through a refined series of a-priori estimates, $^+$0-bounds for all $^+$1 are subsequently constructed for $^+$2, $^+$3, and $^+$4. The approach is an extension of the strategy from the scalar, single-species weakly singular Keller-Segel system [le2025absence] to the coupled CD4$^+$5/CD8$^+$6/IFN-$^+$7 AA context, facing additional difficulties due to the nonlinear coupling term $^+$8.

Formally: $^+$9 for all $v$0, with $v$1 independent of time.

Asymptotic Convergence to Steady State

Beyond boundedness, the analysis demonstrates exponential convergence to the spatially homogeneous steady state

$v$2

where $v$3, under the critical regime $v$4 and $v$5. A Lyapunov functional of entropy type is constructed to facilitate dissipation estimates: $v$6 with

$v$7

where $v$8 is the squared $v$9-distance to steady state. This yields

$\gamma$0

for some $\gamma$1.

The structure of the proof handles the complicated cross-diffusive nonlinearity, controlling all higher-order derivatives by bootstrapping $\gamma$2 and $\gamma$3 bounds, yielding uniform parabolic regularity and ultimately Hölder and Sobolev bounds for all variables.

Numerical Simulation and Spatio-Temporal Dynamics

To corroborate the theoretical predictions, a finite-difference and finite-volume numerical framework is implemented on a two-dimensional domain, with explicit Euler time-stepping and positivity-preserving cutoffs. The system is initialized near the steady state with small random perturbations, and the evolution of $\gamma$4 is visualized at selected times.

The simulations elucidate the dissipative dynamics: transient spatio-temporal patterning (Turing-like instability), subsequent peak coalescence, smoothing towards quasi-uniformity, and eventual sharp convergence towards the homogeneous steady state.

Figure 1: Spatio-temporal evolution and Turing instability of $\gamma$5 for the chemotaxis system with weakly singular sensitivity ($\gamma$6), illustrating early pattern formation and gradual return to equilibrium.

Figure 2: Continued evolution with $\gamma$7, showing the decay and smoothing of spatial heterogeneities over time.

Figure 3: Spatio-temporal evolution for the case $\gamma$8, corresponding to the standard singular sensitivity case; pattern formation is more pronounced, and convergence is slower.

Figure 4: Long-time behavior for $\gamma$9, emphasizing persistent heterogeneity and slower stabilization compared to $w$0.

The computational experiments confirm that weakly singular sensitivity slows the asymptotic convergence—requiring longer times to reach the homogeneous steady state compared to the fully singular ($w$1) or regular case.

Theoretical and Practical Implications

On the theoretical side, the paper resolves a previously open case for AA models: global well-posedness and exponential stabilization for two-species, one-signal systems with weakly singular chemotactic sensitivity. The results extend and refine existing theory on Keller-Segel-type systems and apply directly to higher-structure AA models relevant in biomedical contexts. The methods—particularly the construction and manipulation of tailored energy functionals—are potent for a broad range of multicomponent reaction-diffusion-chemotaxis systems with singular or near-singular structure.

Practically, these findings imply that chemotactic aggregation induced by weakly singular IFN-$w$2 sensitivity cannot produce pathological blow-up or pattern persistence in 2D tissue domains, provided appropriate dissipative mechanisms (logistic-type damping, sufficiently large $w$3) are in effect. For AA, this supports the mathematical well-posedness of spatio-temporal models incorporating more neurobiologically plausible signaling kinetics.

Future Directions

Potential extensions include analysis in higher spatial dimensions ($w$4), fully parabolic cross-diffusion with additional nonlinearities, generalization to signal-dependent diffusion, or inclusion of more biologically faithful coupled signaling and immune dynamics. The perturbation and functional methods employed here could be adapted to infer threshold bounds for singularity exponents $w$5 and to analyze metastable pattern formation for small but nonzero $w$6.

Conclusion

This work establishes global boundedness and exponential convergence for the AA chemotaxis system with weakly singular chemotactic response. The results fill an essential gap in the PDE theory of such coupling-structured autoimmune disease models, providing rigorous analytical guarantees as well as computational evidence for the predicted long-time dynamics.

(2604.26905)