Rigidity for the heat equation with density on Riemannian manifolds through a conformal change

We investigate uniqueness of solution to the heat equation with a density ρ on complete, non-compact weighted Riemannian manifolds of infinite volume. Our main goal is to identify sufficient conditions under which the solution u vanishes identically, assuming that u belongs to a certain weighted Lebesgue space with exponential or polynomial weight, Lᵖφ . We distinguish between the cases p > 1 and $p = 1$ which required stronger assumptions on the manifold and the density function ρ . We develop a unified method based on a conformal transformation of the metric, which allows us to reduce the problem to a standard heat equation on a suitably weighted manifold. In addition, we construct explicit counterexamples on model manifolds which demonstrate optimality of our assumptions on the density ρ .