Long-term behaviour of the CAO-system: Optimal polynomial $H^1$-convergence to degenerate equilibria

Abstract: This article is concerned with a definitive analysis of its long-term dynamics of the CAO-system. We establish the convergence of global strong solutions to constant equilibria in the full $H<sup>1$-topology in the non-perturbative regime. Remarkably, we demonstrate that the CAO-system exhibits a polynomial rate of decay and we determine the optimal rate. This phenomenon stands in stark contrast to the exponential convergence typically expected for uniformly parabolic systems on bounded domains. This algebraic slowing is shown to be a purely nonlinear effect, driven by the emergence of a slow manifold (center manifold) induced by nonlinear coupling conditions. To the best of our knowledge, this provides the first instance of a global polynomial decay result for a non-gradient, non-local dissipative system on a bounded domain.