@article{ASNSP_2001_4_30_1_171_0,
author = {Ishige, Kazuhiro and Murata, Minoru},
title = {Uniqueness of nonnegative solutions of the {Cauchy} problem for parabolic equations on manifolds or domains},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {171--223},
year = {2001},
publisher = {Scuola normale superiore},
volume = {Ser. 4, 30},
number = {1},
mrnumber = {1882029},
zbl = {1024.35010},
language = {en},
url = {https://www.numdam.org/item/ASNSP_2001_4_30_1_171_0/}
}
TY - JOUR AU - Ishige, Kazuhiro AU - Murata, Minoru TI - Uniqueness of nonnegative solutions of the Cauchy problem for parabolic equations on manifolds or domains JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2001 SP - 171 EP - 223 VL - 30 IS - 1 PB - Scuola normale superiore UR - https://www.numdam.org/item/ASNSP_2001_4_30_1_171_0/ LA - en ID - ASNSP_2001_4_30_1_171_0 ER -
%0 Journal Article %A Ishige, Kazuhiro %A Murata, Minoru %T Uniqueness of nonnegative solutions of the Cauchy problem for parabolic equations on manifolds or domains %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2001 %P 171-223 %V 30 %N 1 %I Scuola normale superiore %U https://www.numdam.org/item/ASNSP_2001_4_30_1_171_0/ %G en %F ASNSP_2001_4_30_1_171_0
Ishige, Kazuhiro; Murata, Minoru. Uniqueness of nonnegative solutions of the Cauchy problem for parabolic equations on manifolds or domains. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 30 (2001) no. 1, pp. 171-223. https://www.numdam.org/item/ASNSP_2001_4_30_1_171_0/
[Ai] , Norm estimate of Green operator, perturbation of Green function and integrability of super harmonic functions, Math. Ann. 312 (1998), 289-318. | Zbl | MR
[AM] - , Generalized Cranston-McConnell inequalities and Martin boundaries of unbounded domains, J. Analyse Math. 69 (1996), 137-152. | Zbl | MR
[An1] , On strong barriers and an inequality of Hardy for domains in RN, J. London Math. Soc. 34 (1986), 274-290. | Zbl | MR
[An2] , Negatively curved manifolds, elliptic operators, and the Martin boundary, Ann. of Math. 125 (1987), 495-536. | Zbl | MR
[An3] , First eigenvalues and comparison of Green's functions for elliptic operators on manifolds or domains, J. Analyse Math. 72 (1997), 45-92. | Zbl | MR
[Ara] , Degenerate elliptic operators, Hardy spaces and diffusions on strongly pseudoconvex domains, Tohoku Math. J. 46 (1994), 469-498. | Zbl | MR
[AT] - , 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. | Zbl | MR
[Aro] , Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1968), 607-694. | Zbl | MR | Numdam
[AB1] - , Uniqueness of solutions of the Cauchy problem for parabolic equations, J. Math. Anal. Appl. 13 (1966), 516-526. | Zbl | MR
[AB2] - , Uniqueness of positive solutions of parabolic equations with unbounded coefficients, Colloq. Math. 18 (1967), 126-135. | Zbl | MR
[AS] - , Local behavior of solutions of quasilinear parabolic equations, Arch. Rational Mech. Anal. 25 (1967), 81-122. | Zbl | MR
[Az] , Behavior of diffusion semi-groups at infinity, Bull. Soc. Math. France 102 (1974), 193-240. | Zbl | MR | Numdam
[BM] - , A Saint-Venant principle for Dirichlet forms on discontinuous media, Ann. Mat. Pura Appl.(4) 169 (1995), 125-181. | Zbl | MR
[CS1] - , Degenerate parabolic equations and Harnack inequality, Ann. Mat. Pura Appl. (4) (1983), 139-162. | Zbl | MR
[CS2] - , A Harnack inequalityfordegenerate parabolic equations, Comm. Partial Differential Equations 9 (1984), 719-749. | Zbl | MR
[CS3] - , A remark on a Harnack inequality for degenerate parabolic equations, Rend. Sem. Mat. Univ. Padova 73 (1985), 179-190. | Zbl | MR | Numdam
[CW] - , Harnack's inequality and mean-value inequalities for solutions of degenerate elliptic equations, Comm. Partial Differential Equations 11 (1986), 1111-1134. | Zbl | MR
[D1] , L1 properties of second order elliptic operators, Bull. London Math. Soc. 17 (1985), 417-436. | Zbl | MR
[D2] , "Heat Kernels and Spectral Theory, Cambridge Univ. Press, Cambridge, 1989. | Zbl | MR
[Dod] , Maximum principle for parabolic inequalities and the heat flow on open manifolds, Indiana Univ. Math. J. 32 (1983), 703-716. | Zbl | MR
[Don] , Uniqueness of the positive solutions of the heat equation, Proc. Amer. Math. Soc. 99 (1987), 353-356. | Zbl | MR
[EK] - , Thefiltration equation in a class of functions decreasing at infinity, Proc. Amer. Math. Soc.120 (1994), 825-830. | Zbl | MR
[EG] - , "Measure Theory and Fine Properties of Functions", CRC Press, Boca Raton, 1992. | Zbl | MR
[FS] - , A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash, Arch. Rational Mech. Anal. 96 (1986), 327-338. | Zbl | MR
[Fe] , The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 1-65. | Zbl | MR
[FOT] - - , "Dirichlet Forms and Symmetric Markov Processes, Walter de Gruyter, Berlin, 1994. | Zbl | MR
[Gr1] , On stochastically complete manifolds, Soviet Math. Dokl. 34 (1987), 310-313. | Zbl | MR
[Gr2] , The heat equation on noncompact Riemannian manifolds, Math. USSR Sbornik 72 (1992), 47-77. | Zbl | MR
[Gu] , On the uniform stabilization of solutions of the second mixed problem for a parabolic equation, Math. USSR Sbornik 47 (1984), 439-498. | Zbl
[GW1] - , Mean value and Harnack inequalities for degenerate parabolic equations, Colloquium Math., dedicated to A. Zygmund LX/LXI (1990), 157-194. | Zbl | MR
[GW2] - , Harnack's inequality for degenerate parabolic equations, Comm. Partial Differential Equations 16 (1991), 745-770. | Zbl | MR
[I] , On the behavior of the solutions of degenerate parabolic equations, Nagoya Math. J. 155 (1999), 1-26. | Zbl | MR
[IKO] - - , Linear equations of the second order of parabolic type., Russian Math. Surveys 17 (1972), 1-144.
[IM] - , An intrinsic metric approach to uniqueness of the positive Cauchy problem for parabolic equations, Math. Z. 227 (1998), 313-335. | Zbl | MR
[Kh] , 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. | Zbl | MR
[Kl] , 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. | Zbl | MR
[KT] - , Minimal solutions of the heat equations and uniqueness of the positive Cauchy problem on homogeneous spaces, Proc. Amer. Math. Soc. 94 (1985), 273-278. | Zbl | MR
[LM] - , "Non-Homogeneous Boundary Value Problems and Applications", Vol.I, Springer-Verlag, Berlin-Heidelberg -New York, 1972. | Zbl | MR
[LP] - , Manifolds with group actions and elliptic operators, Memoirs Amer. Math. Soc.112 (1994), no. 540. | Zbl | MR
[LY] - , On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), 153-201. | Zbl | MR
[Mo] , A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101-134. | Zbl | MR
[M1] , 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. | Zbl | MR
[M2] , Non-uniqueness of the positive Cauchy problem for parabolic equations, J. Differential Equations 123 (1995), 343-387. | Zbl | MR
[M3] , 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. | Zbl | MR
[M4] , Uniqueness and non-uniqueness of the positive Cauchy problem for the heat equation on Riemannian manifolds, Proc. Amer. Math. Soc. 123 (1995), 1923-1932. | Zbl | MR
[M5] , Non-uniqueness of the positive Dirichlet problem for parabolic equations in cylinders, J. Func. Anal. 135 (1996), 456-487. | Zbl | MR
[M6] , Semismall perturbations in the Martin theory for elliptic equations, Israel J. Math. 102 (1997), 29-60. | Zbl | MR
[PS] - , Elliptic operators preserving probability, Theory Prob. Appl. 32 (1987), 718-721. | Zbl | MR
[Pinc] , On uniqueness and nonuniqueness of positive Cauchy problem forparabolic equations with unbounded coefficients, Math. Z. 233 (1996), 569-586. | Zbl | MR
[Pins] , "Positive Harmonic Functions and Diffusion", Cambridge Univ. Press, Cambridge, 1995. | Zbl | MR
[Sa1] , Uniformly elliptic operators on Riemannian manifolds, J. Differential Geom. 36 (1992), 417-450. | Zbl | MR
[Sa2] , A note on Poincaré, Sobolev, and Harnack inequality, Duke Math. J. 2 (1992), 27-38. | Zbl | MR
[Sa3] , Parabolic Harnacck inequality for divergence form second order differential operators, Potential Anal. 4 (1995), 429-467. | Zbl | MR
[Stu1] , Analysis on local Dirichlet spaces-I. Recurrence, conservativeness and LP -Liouville properties, J. Reine Angew. Math. 456 (1994), 173-196. | Zbl | MR
[Stu2] STURM, Analysis on local Dirichlet spaces-II. Upper Gaussian estimates for the fundamental solutions of parabolic equations, Osaka J. Math. 32 (1995), 275-312. | Zbl | MR
[Stu3] , Analysis on local Dirichlet spaces-III. Poincaré and parabolic Harnack inequality, J. Math. Pures Appl. (9) 75 (1996), 273-297. | Zbl | MR
[Stu4] , 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. | Zbl | MR
[T] , 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. | Zbl | JFM
[W] , Positive temperatures on an infinite rod, Trans. Amer. Math. Soc. 55 (1944), 85-95. | Zbl | MR





