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

Murata, Minoru

doi:10.1090/S0002-9939-1995-1242097-3

Uniqueness and nonuniqueness of the positive Cauchy problem for the heat equation on Riemannian manifolds
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Proc. Amer. Math. Soc. 123 (1995), 1923-1932 Request permission

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 $- 1$ 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.

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