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

Author: Minoru Murata
Journal: Proc. Amer. Math. Soc. 123 (1995), 1923-1932
MSC: Primary 58G11; Secondary 35K05, 58G30
DOI: https://doi.org/10.1090/S0002-9939-1995-1242097-3
MathSciNet review: 1242097
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 22 (1968), 607–694. MR 435594
A. Ancona and J. C. Taylor, Some remarks on Widder’s theorem and uniqueness of isolated singularities for parabolic equations, Partial differential equations with minimal smoothness and applications (Chicago, IL, 1990) IMA Vol. Math. Appl., vol. 42, Springer, New York, 1992, pp. 15–23. MR 1155849, DOI https://doi.org/10.1007/978-1-4612-2898-1_2
Robert Azencott, Behavior of diffusion semi-groups at infinity, Bull. Soc. Math. France 102 (1974), 193–240. MR 356254
R. L. Bishop and B. O’Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1–49. MR 251664, DOI https://doi.org/10.1090/S0002-9947-1969-0251664-4
Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
Hyeong In Choi, Asymptotic Dirichlet problems for harmonic functions on Riemannian manifolds, Trans. Amer. Math. Soc. 281 (1984), no. 2, 691–716. MR 722769, DOI https://doi.org/10.1090/S0002-9947-1984-0722769-4
E. B. Davies, $L^1$ properties of second order elliptic operators, Bull. London Math. Soc. 17 (1985), no. 5, 417–436. MR 806008, DOI https://doi.org/10.1112/blms/17.5.417
E. B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1989. MR 990239
Jozef Dodziuk, Maximum principle for parabolic inequalities and the heat flow on open manifolds, Indiana Univ. Math. J. 32 (1983), no. 5, 703–716. MR 711862, DOI https://doi.org/10.1512/iumj.1983.32.32046
Harold Donnelly, Uniqueness of positive solutions of the heat equation, Proc. Amer. Math. Soc. 99 (1987), no. 2, 353–356. MR 870800, DOI https://doi.org/10.1090/S0002-9939-1987-0870800-6
Alexandre Freire, On the Martin boundary of Riemannian products, J. Differential Geom. 33 (1991), no. 1, 215–232. MR 1085140
Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
A. A. Grigor′yan, Stochastically complete manifolds, Dokl. Akad. Nauk SSSR 290 (1986), no. 3, 534–537 (Russian). MR 860324
Atsushi Kasue, A Laplacian comparison theorem and function theoretic properties of a complete Riemannian manifold, Japan. J. Math. (N.S.) 8 (1982), no. 2, 309–341. MR 722530, DOI https://doi.org/10.4099/math1924.8.309
R. Z. Has′minskiĭ, Ergodic properties of recurrent diffusion processes and stabilization of the solution of the Cauchy problem for parabolic equations, Teor. Verojatnost. i Primenen. 5 (1960), 196–214 (Russian, with English summary). MR 0133871

L. Karp and P. Li, The heat equation on complete Riemannian manifolds, unpublished.

A. Korányi and J. C. Taylor, Minimal solutions of the heat equation and uniqueness of the positive Cauchy problem on homogeneous spaces, Proc. Amer. Math. Soc. 94 (1985), no. 2, 273–278. MR 784178, DOI https://doi.org/10.1090/S0002-9939-1985-0784178-8
Peter Li and Shing-Tung Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), no. 3-4, 153–201. MR 834612, DOI https://doi.org/10.1007/BF02399203
Bernard Mair and J. C. Taylor, Integral representation of positive solutions of the heat equation, Théorie du potentiel (Orsay, 1983) Lecture Notes in Math., vol. 1096, Springer, Berlin, 1984, pp. 419–433. MR 890370, DOI https://doi.org/10.1007/BFb0100123
Minoru Murata, On construction of Martin boundaries for second order elliptic equations, Publ. Res. Inst. Math. Sci. 26 (1990), no. 4, 585–627. MR 1081506, DOI https://doi.org/10.2977/prims/1195170848
Minoru Murata, Positive harmonic functions on rotationary symmetric Riemannian manifolds, Potential theory (Nagoya, 1990) de Gruyter, Berlin, 1992, pp. 251–259. MR 1167241
Minoru Murata, Uniform restricted parabolic Harnack inequality, separation principle, and ultracontractivity for parabolic equations, Functional analysis and related topics, 1991 (Kyoto), Lecture Notes in Math., vol. 1540, Springer, Berlin, 1993, pp. 277–288. MR 1225823, DOI https://doi.org/10.1007/BFb0085486
Minoru Murata, Non-uniqueness of the positive Cauchy problem for parabolic equations, J. Differential Equations 123 (1995), no. 2, 343–387. MR 1362880, DOI https://doi.org/10.1006/jdeq.1995.1167

---, Sufficient condition for non-uniqueness of the positive Cauchy problem for parabolic equations, Spectral and Scattering Theory and Applications (K. Yajima, ed.), Adv. Stud. Pure Math., vol. 23, Kinokuniya, Tokyo, 1994, pp. 275-282.

Takeyuki Nagasawa, Uniqueness and Widder’s theorem for the heat equation on Riemannian manifolds, Geometry and its applications (Yokohama, 1991) World Sci. Publ., River Edge, NJ, 1993, pp. 161–173. MR 1343269
Yehuda Pinchover, Representation theorems for positive solutions of parabolic equations, Proc. Amer. Math. Soc. 104 (1988), no. 2, 507–515. MR 962821, DOI https://doi.org/10.1090/S0002-9939-1988-0962821-0

T. Sakai, Riemannian geometry, Shokabo, Tokyo, 1992. (Japanese)

Noriaki Suzuki, Huygens property of parabolic functions and a uniqueness theorem, Hiroshima Math. J. 19 (1989), no. 2, 355–361. MR 1027939

S. Täcklind, Sur les classes quasianalytiques des solutions des équations aux dérivées partielles du type parabolique, Nova Acta Regiae Soc. Sci. Upsaliensis Ser. IV 10 (1936), 1-57.

D. V. Widder, Positive temperatures on an infinite rod, Trans. Amer. Math. Soc. 55 (1944), 85–95. MR 9795, DOI https://doi.org/10.1090/S0002-9947-1944-0009795-2