Long‑time asymptotics for each component in the two‑species system

Characterize the long‑time asymptotic behavior of each population u_1 and u_2 in the two‑species system ∂_t u_1 − div(u_1^m ∇g ∗ (u_1 + u_2)) = 0 and ∂_t u_2 − div(u_2^m ∇g ∗ (u_1 + u_2)) = 0 on the d‑dimensional torus, determining whether and how each component converges (e.g., to a constant state) and at what rates, beyond the expected convergence of the sum u_1 + u_2 to its spatial average.

Background

For the single‑species model, entropy solutions exhibit exponential convergence to the spatial average under suitable conditions. In the two‑species setting, the coupled dynamics and nonlinear mobilities complicate the analysis. While one expects the sum to relax, the individual components’ asymptotics remain unresolved.

Clarifying component‑wise asymptotics would deepen understanding of multi‑species aggregation‑diffusion phenomena and guide numerical and modeling strategies.