Uniqueness and nonuniqueness of the positive Cauchy problem for the heat equation on Riemannian manifolds

Abstract

We investigate a uniqueness problem of whether a nonnegative solution of the heat equation on a noncompact Riemannian manifold is uniquely determined by its initial data. A sufficient condition for the uniqueness (resp. nonuniqueness) is given in terms of nonintegrability (resp. integrability) at infinity of − 1 - 1 times a negative function by which the Ricci (resp. sectional) curvature of the manifold is bounded from below (resp. above) at infinity. For a class of manifolds, these sufficient conditions yield a simple criterion for the uniqueness.