Long-time asymptotics in global existence regimes

Determine the long-time asymptotic behavior of solutions to the local fourth-order aggregation–diffusion models—both the one-species equation and the two-species system—under the parameter regimes that ensure global existence, including characterization of convergence to steady states, metastability, or other asymptotic profiles.

Background

The paper shows that the energies associated with the models are nonincreasing, and numerical experiments indicate aggregation, metastability, and convergence to steady states in various parameter regimes. However, rigorous characterization of asymptotic behavior remains unavailable.

Understanding long-time dynamics for fourth-order degenerate equations is challenging, and existing tools (e.g., entropy methods, spectral analysis) are complicated by nonconvex energies and multi-species couplings.