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

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 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.

[Ar] D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 607–694. MR 0435594
[AT] 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, https://doi.org/10.1007/978-1-4612-2898-1_2
[Az] Robert Azencott, Behavior of diffusion semi-groups at infinity, Bull. Soc. Math. France 102 (1974), 193–240. MR 0356254
[BO] R. L. Bishop and B. O’Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1–49. MR 0251664, https://doi.org/10.1090/S0002-9947-1969-0251664-4
[Cha] 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
[Cho] Hyeong In Choi, Asymptotic Dirichlet problems for harmonic functions on Riemannian manifolds, Trans. Amer. Math. Soc. 281 (1984), no. 2, 691–716. MR 722769, https://doi.org/10.1090/S0002-9947-1984-0722769-4
[D1] E. B. Davies, 𝐿¹ properties of second order elliptic operators, Bull. London Math. Soc. 17 (1985), no. 5, 417–436. MR 806008, https://doi.org/10.1112/blms/17.5.417
[D2] E. B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1989. MR 990239
[Dod] 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, https://doi.org/10.1512/iumj.1983.32.32046
[Don] Harold Donnelly, Uniqueness of positive solutions of the heat equation, Proc. Amer. Math. Soc. 99 (1987), no. 2, 353–356. MR 870800, https://doi.org/10.1090/S0002-9939-1987-0870800-6
[Fre] Alexandre Freire, On the Martin boundary of Riemannian products, J. Differential Geom. 33 (1991), no. 1, 215–232. MR 1085140
[Fri] Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
[G] A. A. Grigor′yan, Stochastically complete manifolds, Dokl. Akad. Nauk SSSR 290 (1986), no. 3, 534–537 (Russian). MR 860324
[Ka] 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, https://doi.org/10.4099/math1924.8.309
[Kh] 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
[KL] L. Karp and P. Li, The heat equation on complete Riemannian manifolds, unpublished.
[KT] 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, https://doi.org/10.1090/S0002-9939-1985-0784178-8
[LY] 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, https://doi.org/10.1007/BF02399203
[MT] 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, https://doi.org/10.1007/BFb0100123
[M1] 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, https://doi.org/10.2977/prims/1195170848
[M2] Minoru Murata, Positive harmonic functions on rotationary symmetric Riemannian manifolds, Potential theory (Nagoya, 1990) de Gruyter, Berlin, 1992, pp. 251–259. MR 1167241
[M3] 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, https://doi.org/10.1007/BFb0085486
[M4] Minoru Murata, Non-uniqueness of the positive Cauchy problem for parabolic equations, J. Differential Equations 123 (1995), no. 2, 343–387. MR 1362880, https://doi.org/10.1006/jdeq.1995.1167
[M5] -, 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.
[N] 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
[P] Yehuda Pinchover, Representation theorems for positive solutions of parabolic equations, Proc. Amer. Math. Soc. 104 (1988), no. 2, 507–515. MR 962821, https://doi.org/10.1090/S0002-9939-1988-0962821-0
[Sa] T. Sakai, Riemannian geometry, Shokabo, Tokyo, 1992. (Japanese)
[Su] Noriaki Suzuki, Huygens property of parabolic functions and a uniqueness theorem, Hiroshima Math. J. 19 (1989), no. 2, 355–361. MR 1027939
[T] 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.
[W] D. V. Widder, Positive temperatures on an infinite rod, Trans. Amer. Math. Soc. 55 (1944), 85–95. MR 0009795, https://doi.org/10.1090/S0002-9947-1944-0009795-2