A classical theorem of Repnikov and Èĭdelman establishes that for bounded initial data on Euclidean space, the solution to the heat equation stabilizes pointwise at large times if and only if the ball averages of the initial data converge. While this result extends to manifolds with non-negative Ricci curvature, the behavior alters drastically in negatively curved spaces characterized by exponential volume growth.

In this talk, we study the stabilization problem in the specific setting of Damek-Ricci spaces. We will discuss recent results showing that the convergence of ball averages is a sufficient, but not a necessary, condition for the stabilization of the heat flow. Building on this framework, we establish a new necessary condition for pointwise asymptotic stability on these spaces and provide a counterexample demonstrating that this condition is strictly weaker than the convergence of standard ball averages. This work is based on a joint work with Muna Naik.