Uniqueness of nonnegative solutions of the Cauchy problem for parabolic equations on manifolds or domains
Kazuhiro Ishige; Minoru Murata
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2001)
- Volume: 30, Issue: 1, page 171-223
- ISSN: 0391-173X
Access Full Article
topHow to cite
topReferences
top- [Ai] H. Aikawa, Norm estimate of Green operator, perturbation of Green function and integrability of super harmonic functions, Math. Ann.312 (1998), 289-318. Zbl0917.31001MR1671780
- [AM] H. Aikawa - M. Murata, Generalized Cranston-McConnell inequalities and Martin boundaries of unbounded domains, J. Analyse Math.69 (1996), 137-152. Zbl0865.31009MR1428098
- [An1] A. Ancona, On strong barriers and an inequality of Hardy for domains in RN, J. London Math. Soc.34 (1986), 274-290. Zbl0629.31002MR856511
- [An2] A. Ancona, Negatively curved manifolds, elliptic operators, and the Martin boundary, Ann. of Math.125 (1987), 495-536. Zbl0652.31008MR890161
- [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. Zbl0944.58016MR1482989
- [Ara] H. Arai, Degenerate elliptic operators, Hardy spaces and diffusions on strongly pseudoconvex domains, Tohoku Math. J.46 (1994), 469-498. Zbl0823.32002MR1301285
- [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. Zbl0816.35044MR1155849
- [Aro] D.G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1968), 607-694. Zbl0182.13802MR435594
- [AB1] D.G. Aronson - P. Besala, Uniqueness of solutions of the Cauchy problem for parabolic equations, J. Math. Anal. Appl.13 (1966), 516-526. Zbl0137.29501MR192197
- [AB2] D.G. Aronson - P. Besala, Uniqueness of positive solutions of parabolic equations with unbounded coefficients, Colloq. Math.18 (1967), 126-135. Zbl0157.17404MR219900
- [AS] D.G. Aronson - J. Serrin, Local behavior of solutions of quasilinear parabolic equations, Arch. Rational Mech. Anal.25 (1967), 81-122. Zbl0154.12001MR244638
- [Az] R. Azencott, Behavior of diffusion semi-groups at infinity, Bull. Soc. Math. France102 (1974), 193-240. Zbl0293.60071MR356254
- [BM] M. Biroli - U. Mosco, A Saint-Venant principle for Dirichlet forms on discontinuous media, Ann. Mat. Pura Appl.(4) 169 (1995), 125-181. Zbl0851.31008MR1378473
- [CS1] F. Chiarenza - R. Serapioni, Degenerate parabolic equations and Harnack inequality, Ann. Mat. Pura Appl. (4) (1983), 139-162. Zbl0573.35052MR772255
- [CS2] F. Chiapenza - R. Serapioni, A Harnack inequalityfordegenerate parabolic equations, Comm. Partial Differential Equations9 (1984), 719-749. Zbl0546.35035MR748366
- [CS3] F. Chiarenza - R. Serapioni, A remark on a Harnack inequality for degenerate parabolic equations, Rend. Sem. Mat. Univ. Padova73 (1985), 179-190. Zbl0588.35013MR799906
- [CW] S. Chanillo - R.L. Wheeden, Harnack's inequality and mean-value inequalities for solutions of degenerate elliptic equations, Comm. Partial Differential Equations11 (1986), 1111-1134. Zbl0634.35035MR847996
- [D1] E.B. Davies, L1 properties of second order elliptic operators, Bull. London Math. Soc.17 (1985), 417-436. Zbl0583.35032MR806008
- [D2] E.B. Davies, "Heat Kernels and Spectral Theory, Cambridge Univ. Press, Cambridge, 1989. Zbl0699.35006MR990239
- [Dod] J. Dodziuk, Maximum principle for parabolic inequalities and the heat flow on open manifolds, Indiana Univ. Math. J.32 (1983), 703-716. Zbl0526.58047MR711862
- [Don] H. Donnelly, Uniqueness of the positive solutions of the heat equation, Proc. Amer. Math. Soc.99 (1987), 353-356. Zbl0615.53034MR870800
- [EK] D. Eidus - S. Kamin, Thefiltration equation in a class of functions decreasing at infinity, Proc. Amer. Math. Soc.120 (1994), 825-830. Zbl0791.35065MR1169025
- [EG] L.C. Evans - R.F. Gariepy, "Measure Theory and Fine Properties of Functions", CRC Press, Boca Raton, 1992. Zbl0804.28001MR1158660
- [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. Zbl0652.35052MR855753
- [Fe] C. Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math.26 (1974), 1-65. Zbl0289.32012MR350069
- [FOT] M. Fukushima - Y. Oshima - M. Takeda, "Dirichlet Forms and Symmetric Markov Processes, Walter de Gruyter, Berlin, 1994. Zbl0838.31001MR1303354
- [Gr1] A.A. Grigor'yan, On stochastically complete manifolds, Soviet Math. Dokl.34 (1987), 310-313. Zbl0632.58041MR860324
- [Gr2] A.A. Grigor'yan, The heat equation on noncompact Riemannian manifolds, Math. USSR Sbornik72 (1992), 47-77. Zbl0776.58035MR1098839
- [Gu] A.K. Gushchin, On the uniform stabilization of solutions of the second mixed problem for a parabolic equation, Math. USSR Sbornik47 (1984), 439-498. Zbl0554.35055
- [GW1] C.E. Gutiérrez- R.L. Wheeden, Mean value and Harnack inequalities for degenerate parabolic equations, Colloquium Math., dedicated to A. ZygmundLX/LXI (1990), 157-194. Zbl0785.35057MR1096367
- [GW2] C.E. Gutiérrez - R.L. Wheeden, Harnack's inequality for degenerate parabolic equations, Comm. Partial Differential Equations16 (1991), 745-770. Zbl0746.35007MR1113105
- [I] K. Ishige, On the behavior of the solutions of degenerate parabolic equations, Nagoya Math. J.155 (1999), 1-26. Zbl0932.35131MR1711391
- [IKO] A.M. Il'n - A.S. Kalashnikov - O.A. Oleinik, Linear equations of the second order of parabolic type., Russian Math. Surveys17 (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. Zbl0893.35042MR1609065
- [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. Zbl0093.14902MR133871
- [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. Zbl0422.53032MR463506
- [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. Zbl0577.35047MR784178
- [LM] J.L. Lions — E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications", Vol.I, Springer-Verlag, Berlin-Heidelberg -New York, 1972. Zbl0223.35039MR350177
- [LP] V. Lin - Y. Pinchover, Manifolds with group actions and elliptic operators, Memoirs Amer. Math. Soc.112 (1994), no. 540. Zbl0816.58041MR1230774
- [LY] P. Li - S.T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math.156 (1986), 153-201. Zbl0611.58045MR834612
- [Mo] J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math.17 (1964), 101-134. Zbl0149.06902MR159139
- [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. Zbl0793.35046MR1225823
- [M2] M. Murata, Non-uniqueness of the positive Cauchy problem for parabolic equations, J. Differential Equations123 (1995), 343-387. Zbl0843.35036MR1362880
- [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. Zbl0806.35054MR1275409
- [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. Zbl0829.58042MR1242097
- [M5] M. Murata, Non-uniqueness of the positive Dirichlet problem for parabolic equations in cylinders, J. Func. Anal.135 (1996), 456-487. Zbl0846.35055MR1370610
- [M6] M. Murata, Semismall perturbations in the Martin theory for elliptic equations, Israel J. Math.102 (1997), 29-60. Zbl0891.35013MR1489100
- [PS] M.A. Perel'muter - Yu A. Semenov, Elliptic operators preserving probability, Theory Prob. Appl.32 (1987), 718-721. Zbl0715.35025MR927262
- [Pinc] Y. Pinchover, On uniqueness and nonuniqueness of positive Cauchy problem forparabolic equations with unbounded coefficients, Math. Z.233 (1996), 569-586. Zbl0869.35010MR1421956
- [Pins] R.G. Pinsky, "Positive Harmonic Functions and Diffusion", Cambridge Univ. Press, Cambridge, 1995. Zbl0858.31001MR1326606
- [Sa1] L. Saloff-Coste, Uniformly elliptic operators on Riemannian manifolds, J. Differential Geom.36 (1992), 417-450. Zbl0735.58032MR1180389
- [Sa2] L. Saloff-Coste, A note on Poincaré, Sobolev, and Harnack inequality, Duke Math. J.2 (1992), 27-38. Zbl0769.58054MR1150597
- [Sa3] L. Saloff-Coste, Parabolic Harnacck inequality for divergence form second order differential operators, Potential Anal.4 (1995), 429-467. Zbl0840.31006MR1354894
- [Stu1] K. Th. Sturm, Analysis on local Dirichlet spaces-I. Recurrence, conservativeness and LP -Liouville properties, J. Reine Angew. Math.456 (1994), 173-196. Zbl0806.53041MR1301456
- [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. Zbl0854.35015MR1355744
- [Stu3] K. Th. Sturm, Analysis on local Dirichlet spaces-III. Poincaré and parabolic Harnack inequality, J. Math. Pures Appl. (9) 75 (1996), 273-297. Zbl0854.35016MR1387522
- [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. Zbl0834.58039MR1360279
- [T] Täcklind, Sur les classes quasianalytiques des solutions des équations aux dérivées partielles du type parabolique, Nova Acta Regiae Soc. Scien. Upsaliensis, Ser. IV10 (1936), 1-57. Zbl0014.02204JFM62.1186.01
- [W] D.V. Widder, Positive temperatures on an infinite rod, Trans. Amer. Math. Soc.55 (1944), 85-95. Zbl0061.22303MR9795