Uniqueness of nonnegative solutions of the Cauchy problem for parabolic equations on manifolds or domains

Ishige, Kazuhiro; Murata, Minoru

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 30 (2001) no. 1, pp. 171-223.

[Ai] H. Aikawa, Norm estimate of Green operator, perturbation of Green function and integrability of super harmonic functions, Math. Ann. 312 (1998), 289-318. | MR | Zbl

[AM] H. Aikawa - M. Murata, Generalized Cranston-McConnell inequalities and Martin boundaries of unbounded domains, J. Analyse Math. 69 (1996), 137-152. | MR | Zbl

[An1] A. Ancona, On strong barriers and an inequality of Hardy for domains in RN, J. London Math. Soc. 34 (1986), 274-290. | MR | Zbl

[An2] A. Ancona, Negatively curved manifolds, elliptic operators, and the Martin boundary, Ann. of Math. 125 (1987), 495-536. | MR | Zbl

[An3] A. Ancona, First eigenvalues and comparison of Green's functions for elliptic operators on manifolds or domains, J. Analyse Math. 72 (1997), 45-92. | MR | Zbl

[Ara] H. Arai, Degenerate elliptic operators, Hardy spaces and diffusions on strongly pseudoconvex domains, Tohoku Math. J. 46 (1994), 469-498. | MR | Zbl

[AT] A. Ancona - J.C. Taylor, Some remarks on Widder's theorem and uniqueness of isolated singularities for parabolic equations, In: "Partial Differential Equations with Minimal Smoothness and Applications", B. Dahlberg et al. (eds), Springer-Velag, New York, 1992, pp.15-23. | MR | Zbl

[Aro] D.G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1968), 607-694. | Numdam | MR | Zbl

[AB1] D.G. Aronson - P. Besala, Uniqueness of solutions of the Cauchy problem for parabolic equations, J. Math. Anal. Appl. 13 (1966), 516-526. | MR | Zbl

[AB2] D.G. Aronson - P. Besala, Uniqueness of positive solutions of parabolic equations with unbounded coefficients, Colloq. Math. 18 (1967), 126-135. | MR | Zbl

[AS] D.G. Aronson - J. Serrin, Local behavior of solutions of quasilinear parabolic equations, Arch. Rational Mech. Anal. 25 (1967), 81-122. | MR | Zbl

[Az] R. Azencott, Behavior of diffusion semi-groups at infinity, Bull. Soc. Math. France 102 (1974), 193-240. | Numdam | MR | Zbl

[BM] M. Biroli - U. Mosco, A Saint-Venant principle for Dirichlet forms on discontinuous media, Ann. Mat. Pura Appl.(4) 169 (1995), 125-181. | MR | Zbl

[CS1] F. Chiarenza - R. Serapioni, Degenerate parabolic equations and Harnack inequality, Ann. Mat. Pura Appl. (4) (1983), 139-162. | MR | Zbl

[CS2] F. Chiapenza - R. Serapioni, A Harnack inequalityfordegenerate parabolic equations, Comm. Partial Differential Equations 9 (1984), 719-749. | MR | Zbl

[CS3] F. Chiarenza - R. Serapioni, A remark on a Harnack inequality for degenerate parabolic equations, Rend. Sem. Mat. Univ. Padova 73 (1985), 179-190. | Numdam | MR | Zbl

[CW] S. Chanillo - R.L. Wheeden, Harnack's inequality and mean-value inequalities for solutions of degenerate elliptic equations, Comm. Partial Differential Equations 11 (1986), 1111-1134. | MR | Zbl

[D1] E.B. Davies, L1 properties of second order elliptic operators, Bull. London Math. Soc. 17 (1985), 417-436. | MR | Zbl

[D2] E.B. Davies, "Heat Kernels and Spectral Theory, Cambridge Univ. Press, Cambridge, 1989. | MR | Zbl

[Dod] J. Dodziuk, Maximum principle for parabolic inequalities and the heat flow on open manifolds, Indiana Univ. Math. J. 32 (1983), 703-716. | MR | Zbl

[Don] H. Donnelly, Uniqueness of the positive solutions of the heat equation, Proc. Amer. Math. Soc. 99 (1987), 353-356. | MR | Zbl

[EK] D. Eidus - S. Kamin, Thefiltration equation in a class of functions decreasing at infinity, Proc. Amer. Math. Soc.120 (1994), 825-830. | MR | Zbl

[EG] L.C. Evans - R.F. Gariepy, "Measure Theory and Fine Properties of Functions", CRC Press, Boca Raton, 1992. | MR | Zbl

[FS] E.B. Fabes - D.W. Stroock, A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash, Arch. Rational Mech. Anal. 96 (1986), 327-338. | MR | Zbl

[Fe] C. Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 1-65. | MR | Zbl

[FOT] M. Fukushima - Y. Oshima - M. Takeda, "Dirichlet Forms and Symmetric Markov Processes, Walter de Gruyter, Berlin, 1994. | MR | Zbl

[Gr1] A.A. Grigor'Yan, On stochastically complete manifolds, Soviet Math. Dokl. 34 (1987), 310-313. | MR | Zbl

[Gr2] A.A. Grigor'Yan, The heat equation on noncompact Riemannian manifolds, Math. USSR Sbornik 72 (1992), 47-77. | MR | Zbl

[Gu] A.K. Gushchin, On the uniform stabilization of solutions of the second mixed problem for a parabolic equation, Math. USSR Sbornik 47 (1984), 439-498. | Zbl

[GW1] C.E. Gutiérrez- R.L. Wheeden, Mean value and Harnack inequalities for degenerate parabolic equations, Colloquium Math., dedicated to A. Zygmund LX/LXI (1990), 157-194. | MR | Zbl

[GW2] C.E. Gutiérrez - R.L. Wheeden, Harnack's inequality for degenerate parabolic equations, Comm. Partial Differential Equations 16 (1991), 745-770. | MR | Zbl

[I] K. Ishige, On the behavior of the solutions of degenerate parabolic equations, Nagoya Math. J. 155 (1999), 1-26. | MR | Zbl

[IKO] A.M. Il'N - A.S. Kalashnikov - O.A. Oleinik, Linear equations of the second order of parabolic type., Russian Math. Surveys 17 (1972), 1-144.

[IM] K. Ishige - M. Murata, An intrinsic metric approach to uniqueness of the positive Cauchy problem for parabolic equations, Math. Z. 227 (1998), 313-335. | MR | Zbl

[Kh] R.Z. Khas'Minskii, Ergodic properties of recurrent diffusion processes and stabilization of the solution to the Cauchy problem for parabolic equations, Theory Prob. Appl. 5 (1960), 179-196. | MR | Zbl

[Kl] P.F. Klembeck, Kähler metrics of negative curvature, the Bergman metric near the boundary, and the Kobayashi metric on smooth bounded strictly pseudoconvex sets, Indiana Univ. Math. J. 27 (1978), 275-282. | MR | Zbl

[KT] A. Koranyi - J.C. Taylor, Minimal solutions of the heat equations and uniqueness of the positive Cauchy problem on homogeneous spaces, Proc. Amer. Math. Soc. 94 (1985), 273-278. | MR | Zbl

[LM] J.L. Lions - E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications", Vol.I, Springer-Verlag, Berlin-Heidelberg -New York, 1972. | MR | Zbl

[LP] V. Lin - Y. Pinchover, Manifolds with group actions and elliptic operators, Memoirs Amer. Math. Soc.112 (1994), no. 540. | MR | Zbl

[LY] P. Li - S.T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), 153-201. | MR | Zbl

[Mo] J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101-134. | MR | Zbl

[M1] M. Murata, Uniform restricted parabolic Harnack inequality, separation principle, and ultracontractivity for parabolic equations, In: "Functional Analysis and Related Topics", 1991, Lecture Notes in Math. Vol. 1540, H. Komatsu (ed.), Springer-Verlag, Berlin, 1993, pp. 277-285. | MR | Zbl

[M2] M. Murata, Non-uniqueness of the positive Cauchy problem for parabolic equations, J. Differential Equations 123 (1995), 343-387. | MR | Zbl

[M3] M. Murata, Sufficient condition for non-uniqueness of the positive Cauchy problem for parabolic equations, In: "Spectral and Scattering Theory and Applications", Advanced Studies in Pure Math., Vol. 23, Kinokuniya, Tokyo, 1994, pp. 275-282. | MR | Zbl

[M4] M. Murata, Uniqueness and non-uniqueness of the positive Cauchy problem for the heat equation on Riemannian manifolds, Proc. Amer. Math. Soc. 123 (1995), 1923-1932. | MR | Zbl

[M5] M. Murata, Non-uniqueness of the positive Dirichlet problem for parabolic equations in cylinders, J. Func. Anal. 135 (1996), 456-487. | MR | Zbl

[M6] M. Murata, Semismall perturbations in the Martin theory for elliptic equations, Israel J. Math. 102 (1997), 29-60. | MR | Zbl

[PS] M.A. Perel'Muter - Yu A. Semenov, Elliptic operators preserving probability, Theory Prob. Appl. 32 (1987), 718-721. | MR | Zbl

[Pinc] Y. Pinchover, On uniqueness and nonuniqueness of positive Cauchy problem forparabolic equations with unbounded coefficients, Math. Z. 233 (1996), 569-586. | MR | Zbl

[Pins] R.G. Pinsky, "Positive Harmonic Functions and Diffusion", Cambridge Univ. Press, Cambridge, 1995. | MR | Zbl

[Sa1] L. Saloff-Coste, Uniformly elliptic operators on Riemannian manifolds, J. Differential Geom. 36 (1992), 417-450. | MR | Zbl

[Sa2] L. Saloff-Coste, A note on Poincaré, Sobolev, and Harnack inequality, Duke Math. J. 2 (1992), 27-38. | MR | Zbl

[Sa3] L. Saloff-Coste, Parabolic Harnacck inequality for divergence form second order differential operators, Potential Anal. 4 (1995), 429-467. | MR | Zbl

[Stu1] K. Th. Sturm, Analysis on local Dirichlet spaces-I. Recurrence, conservativeness and LP -Liouville properties, J. Reine Angew. Math. 456 (1994), 173-196. | MR | Zbl

[Stu2] K. Th. STURM, Analysis on local Dirichlet spaces-II. Upper Gaussian estimates for the fundamental solutions of parabolic equations, Osaka J. Math. 32 (1995), 275-312. | MR | Zbl

[Stu3] K. Th. Sturm, Analysis on local Dirichlet spaces-III. Poincaré and parabolic Harnack inequality, J. Math. Pures Appl. (9) 75 (1996), 273-297. | MR | Zbl

[Stu4] K. Th. Sturm, On the geometry defined by Dirichletforms, In: "Seminar on Stochastic Analysis, Random Fields and Applications", E. Bolthansen et al. (eds.) (Progress in Prob. vol. 36), Birkhäuser, 1995, pp. 231-242. | MR | Zbl

[T] Täcklind, Sur les classes quasianalytiques des solutions des équations aux dérivées partielles du type parabolique, Nova Acta Regiae Soc. Scien. Upsaliensis, Ser. IV 10 (1936), 1-57. | JFM | Zbl

[W] D.V. Widder, Positive temperatures on an infinite rod, Trans. Amer. Math. Soc. 55 (1944), 85-95. | MR | Zbl