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.

References

The asymptotic behaviour of each population also remains an open problem, although one expects their sum to converge to its spatial average.

— On a repulsion model with Coulomb interaction and nonlinear mobility  (2510.16894 - Courcel et al., 19 Oct 2025) in Subsection “Possible extensions” (Introduction)

The long-time behavior of solutions is widely an open problem.

— Superlinear nonlocal diffusion systems for multispecies populations: well-posedness and discrete chain rules  (2609.09921 - Hirvonen et al., 9 Sep 2026) in Section 1, subsection “Main results”; see also Remark 2.14, “Long-time asymptotics”

The long-time behavior for $1<s<2$ is an open problem, since we exploit heavily the fact that $z\mapsto z{2/s}$ is H"older continuous.

— Superlinear nonlocal diffusion systems for multispecies populations: well-posedness and discrete chain rules  (2609.09921 - Hirvonen et al., 9 Sep 2026) in Commented-out Section “Long-time behavior of solutions”