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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

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


Author: Minoru Murata
Journal: Proc. Amer. Math. Soc. 123 (1995), 1923-1932
MSC: Primary 58G11; Secondary 35K05, 58G30
MathSciNet review: 1242097
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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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DOI: https://doi.org/10.1090/S0002-9939-1995-1242097-3
Keywords: Uniqueness, positive Cauchy problem, heat equation, Riemannian manifold
Article copyright: © Copyright 1995 American Mathematical Society