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Volumn 30, Issue 1, 2001, Pages 171-223

Uniqueness of Nonnegative Solutions of the Cauchy Problem for Parabolic Equations on Manifolds or Domains

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EID: 0000738895     PISSN: 0391173X     EISSN: 20362145     Source Type: Journal    
DOI: None     Document Type: Article
Times cited : (54)

References (59)
  • 1
    • 0032329262 scopus 로고    scopus 로고
    • Norm estimate of Green operator, perturbation of Green function and integrability of superharmonic functions
    • [Ai]
    • [Ai] H. AïKAWA, Norm estimate of Green operator, perturbation of Green function and integrability of superharmonic functions. Math. Ann. 312 (1998), 289-318.
    • (1998) Math. Ann , vol.312 , pp. 289-318
    • AïKAWA, H.1
  • 2
    • 0000801374 scopus 로고    scopus 로고
    • Generalized Cranston-McConnell inequalities and Martin boundaries of unbounded domains
    • [AM]
    • [AM] H . A i K A W A - M . MURATA, Generalized Cranston-McConnell inequalities and Martin boundaries of unbounded domains, J. Analyse Math. 69 (1996), 137-152.
    • (1996) J. Analyse Math , vol.69 , pp. 137-152
    • MURATA, H . A i K A W A - M .1
  • 3
    • 84963015761 scopus 로고
    • On strong barriers andan inequalityofHardyfordomainsinR^
    • [Ani]
    • [Ani] A. ANCONA, On strong barriers andan inequalityofHardyfordomainsinR^, J. London Math. Soc. 34 (1986), 274-290.
    • (1986) J. London Math. Soc , vol.34 , pp. 274-290
    • ANCONA, A.1
  • 4
    • 0000126969 scopus 로고
    • Negatively curved manifolds, elliptic operators, and the Martin boundary
    • [An2]
    • [An2] A . ANCONA, Negatively curved manifolds, elliptic operators, and the Martin boundary, Ann. of Math. 125 (1987), 495-536.
    • (1987) Ann. of Math , vol.125 , pp. 495-536
    • ANCONA, A .1
  • 5
    • 0002417396 scopus 로고    scopus 로고
    • First eigenvalues and comparison of Green *s functions for elliptic operators on manifolds or domains
    • [An3]
    • [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.
    • (1997) J. Analyse Math , vol.72 , pp. 45-92
    • ANCONA, A.1
  • 6
    • 84972573730 scopus 로고
    • Degenerate elliptic operators. Hardy spaces and diffusions on strongly pseudoconvex domains
    • [Ara] H A R A I
    • [Ara] H . A R A I, Degenerate elliptic operators. Hardy spaces and diffusions on strongly pseudoconvex domains, Tohoku Math. J. 46 (1994), 469-498.
    • (1994) Tohoku Math. J , vol.46 , pp. 469-498
  • 7
    • 0002259005 scopus 로고
    • Some remarks on Widderts theorem and uniqueness of isolated singularities for parabolic equations
    • [AT] B. Dahlberg et al. (eds). Springer-Velag, New York
    • [AT] A . ANCONA - J . C. TAYLOR, Some remarks on Widderts 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.
    • (1992) Partial Differential Equations with Minimal Smoothness and Applications , pp. 15-23
    • ANCONA, A .1    TAYLOR, J . C.2
  • 8
    • 0000377165 scopus 로고
    • Non-negative solutions oflinear parabolic equations
    • [Aro]
    • [Aro] D. G. ARONSON, Non-negative solutions oflinear parabolic equations, Ann. Scuola Norm. Sup. Pisa CI. Sci. (4) 22 (1968), 607-694.
    • (1968) Ann. Scuola Norm. Sup. Pisa CI. Sci. (4) , vol.22 , pp. 607-694
    • ARONSON, D. G.1
  • 9
    • 0041115324 scopus 로고
    • Uniqueness of solutions of the Cauchy problem for parabolic equations
    • [ABl]
    • [ABl] D. G. ARONSON - P. BÉSALA, Uniqueness of solutions of the Cauchy problem for parabolic equations, J. Math. Anal. Appi. 13 (1966), 516-526.
    • (1966) J. Math. Anal. Appi , vol.13 , pp. 516-526
    • ARONSON, D. G.1    BÉSALA, P.2
  • 10
    • 0009554263 scopus 로고
    • Uniqueness of positive solutions of parabolic equations with unbounded coefficients
    • [AB2]
    • [AB2] D. G. ARONSON - P. BÉSALA, Uniqueness of positive solutions of parabolic equations with unbounded coefficients, Colloq. Math. 18 (1967), 126-135.
    • (1967) Colloq. Math , vol.18 , pp. 126-135
    • ARONSON, D. G.1    BÉSALA, P.2
  • 12
    • 0002524069 scopus 로고
    • Behavior of diffusion semi-groups at infinity
    • [Az]
    • [Az] R. AzENCOTT, Behavior of diffusion semi-groups at infinity. Bull. Soc. Math. France 102 (1974), 193-240.
    • (1974) Bull. Soc. Math. France , vol.102 , pp. 193-240
    • AzENCOTT, R.1
  • 13
    • 0000067861 scopus 로고
    • A Saint-Venant principle for Dirichlet forms on discontinuous media
    • [BM]
    • [BM] M. BiROLi - U. Mosco, A Saint-Venant principle for Dirichlet forms on discontinuous media, Ann. Mat. Pura Appl.(4) 169 (1995), 125-181.
    • (1995) Ann. Mat. Pura Appl , vol.169 , Issue.4 , pp. 125-181
    • BiROLi, M.1    Mosco, U.2
  • 14
    • 0010072199 scopus 로고
    • Degenerate parabolic equations and Hamack inequality
    • [CS 1]
    • [CS 1] F. CHIARENZA - R. SERAHONI, Degenerate parabolic equations and Hamack inequality, Ann. Mat. Pura Appi. (4) (1983), 139-162.
    • (1983) Ann. Mat. Pura Appi , Issue.4 , pp. 139-162
    • CHIARENZA, F.1    SERAHONI, R.2
  • 16
    • 0001217124 scopus 로고
    • A remark on a Hamack inequality for degenerate parabolic equations
    • [CS3]
    • [CS3] F . CHIARENZA - R. SERAPIONI, A remark on a Hamack inequality for degenerate parabolic equations. Rend. Sem. Mat. Univ. Padova 73 (1985), 179-190.
    • (1985) Rend. Sem. Mat. Univ. Padova , vol.73 , pp. 179-190
    • CHIARENZA, F .1    SERAPIONI, R.2
  • 17
    • 0001763939 scopus 로고
    • Hamack's inequality and mean-value inequalities for solutions of degenerate elliptic equations
    • [CW]
    • [CW] S. CHANILLO - R . L . WHEEDEN, Hamack's inequality and mean-value inequalities for solutions of degenerate elliptic equations. Comm. Partial Differential Equations 11 (1986), 1111-1134.
    • (1986) Comm. Partial Differential Equations , vol.11 , pp. 1111-1134
    • CHANILLO, S.1    WHEEDEN, R . L .2
  • 18
    • 0001201473 scopus 로고
    • properties of second order elliptic operators
    • [D l]
    • [ D l ] E . B . D A VIES, properties of second order elliptic operators. Bull. London Math. Soc. 17 (1985), 417-436.
    • (1985) Bull. London Math. Soc , vol.17 , pp. 417-436
    • VIES, E . B . D A1
  • 20
    • 0001921997 scopus 로고
    • Maximum principle for parabolic inequalities and the heat flow on open manifolds
    • [Dod]
    • [Dod] J . DODZIUK, Maximum principle for parabolic inequalities and the heat flow on open manifolds, Indiana Univ. Math. J. 32 (1983), 703-716.
    • (1983) Indiana Univ. Math. J , vol.32 , pp. 703-716
    • DODZIUK, J .1
  • 21
    • 84968465022 scopus 로고
    • Uniqueness of the positive solutions of the heat equation
    • [Don]
    • [Don] H . DONNELLY, Uniqueness of the positive solutions of the heat equation, Proc. Amer. Math. Soc. 99 (1987), 353-356.
    • (1987) Proc. Amer. Math. Soc , vol.99 , pp. 353-356
    • DONNELLY, H .1
  • 22
    • 84966225790 scopus 로고
    • Thefiltration equation in a class of functions decreasing at infinity
    • [EK]
    • [EK] D. EiDUS - S. KAMIN, Thefiltration equation in a class of functions decreasing at infinity, Proc. Amer. Math. Soc. 120 (1994), 825-830.
    • (1994) Proc. Amer. Math. Soc , vol.120 , pp. 825-830
    • EiDUS, D.1    KAMIN, S.2
  • 24
    • 33646917618 scopus 로고
    • A new proof of Mosens parabolic Hamack inequality using the old ideas of Nash
    • [FS]
    • [FS] E . B . FABES - D. W. STROOCK, A new proof of Mosens parabolic Hamack inequality using the old ideas of Nash, Arch. Rational Mech. Anal. 96 (1986), 327-338.
    • (1986) Arch. Rational Mech. Anal , vol.96 , pp. 327-338
    • FABES, E . B .1    STROOCK, D. W.2
  • 27
    • 0000070043 scopus 로고
    • On stochastically complete manifolds
    • [Grl]
    • [Grl] A. A. GRICX)R'YAN, On stochastically complete manifolds, Soviet Math. Dokl. 34 (1987), 310-313.
    • (1987) Soviet Math. Dokl , vol.34 , pp. 310-313
    • GRICX, A. A.1    R'YAN2
  • 28
    • 84956103754 scopus 로고
    • The heat equation on noncompact Riemannian manifolds
    • [Gr2]
    • [Gr2] A. A. GRIGOR'YAN, The heat equation on noncompact Riemannian manifolds. Math. USSR Sbomik 72 (1992), 47-77.
    • (1992) Math. USSR Sbomik , vol.72 , pp. 47-77
    • GRIGOR'YAN, A. A.1
  • 29
    • 84956225532 scopus 로고
    • On the uniform stabilization of solutions of the second mixed problem for a parabolic equation
    • [Gu]
    • [Gu] A. K. GusHCHiN, On the uniform stabilization of solutions of the second mixed problem for a parabolic equation. Math. USSR Sbomik 47 (1984), 439-498.
    • (1984) Math. USSR Sbomik , vol.47 , pp. 439-498
    • GusHCHiN, A. K.1
  • 32
    • 0002155550 scopus 로고    scopus 로고
    • On the behavior of the solutions of degenerate parabolic equations
    • [I]
    • [I] K . ISHIGE, On the behavior of the solutions of degenerate parabolic equations, Nagoya Math. J. 155(1999), 1-26.
    • (1999) Nagoya Math. J , vol.155 , pp. 1-26
    • ISHIGE, K .1
  • 33
    • 0002035256 scopus 로고
    • Linear equations of the second order of parabolic type
    • [IKO] A. M. I L N A
    • [IKO] A. M. I L ' N - A . S. KALASHNIKOV - O. A. OvEmm, Linear equations of the second order of parabolic type, Russian Math. Surveys 17 (1972), 1-144.
    • (1972) Russian Math. Surveys , vol.17 , pp. 1-144
    • KALASHNIKOV, S.1    OvEmm, O. A.2
  • 34
    • 0032346940 scopus 로고    scopus 로고
    • An intrinsic metric approach to uniqueness of the positive Cauchy problem for parabolic equations
    • [IM] M M U R A T A
    • [IM] K . IsHiGE - M . M U R A T A, An intrinsic metric approach to uniqueness of the positive Cauchy problem for parabolic equations. Math. Z. 227 (1998), 313-335.
    • (1998) Math. Z , vol.227 , pp. 313-335
    • IsHiGE, K .1
  • 35
    • 0000073325 scopus 로고
    • Ergodic properties of recurrent diffusion processes and stabilization of the solution to the Cauchy problem for parabolic equations
    • [Kh]
    • [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. Appi. 5 (1960), 179-196.
    • (1960) Theory Prob. Appi , vol.5 , pp. 179-196
    • KHAS'MINSKII, R. Z.1
  • 36
    • 0000116541 scopus 로고
    • Kahler metrics of negative curvature, the Bergman metric near the boundary, and the Kobayashi metric on smooth bounded strictly pseudoconvex sets
    • [Kl]
    • [Kl] P. F . KLEMBECK, Kahler 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.
    • (1978) Indiana Univ. Math. J , vol.27 , pp. 275-282
    • KLEMBECK, P. F .1
  • 37
    • 84968518028 scopus 로고
    • Minimal solutions of the heat equations and uniqueness of the positive Cauchy problem on homogeneous spaces
    • Y i [KT]
    • [KT] A. KORAN Y i - 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.
    • (1985) Proc. Amer. Math. Soc , vol.94 , pp. 273-278
    • KORAN, A.1    TAYLOR, J. C.2
  • 39
    • 0002630285 scopus 로고
    • Manifolds with group actions and elliptic operators
    • [LP]
    • [LP] V. L I N - Y. PINCHOVER, Manifolds with group actions and elliptic operators. Memoirs Amer. Math. Soc. 112 (1994), no. 540.
    • (1994) Memoirs Amer. Math. Soc , vol.112 , Issue.540
    • PINCHOVER, V. L I N - Y.1
  • 40
    • 0000324115 scopus 로고
    • On the parabolic kernel of the Schrödinger operator
    • S. T. Y A U
    • P. LI - S. T. Y A U, On the parabolic kernel of the Schrödinger operator. Acta Math. 156(1986), 153-201.
    • (1986) Acta Math , vol.156 , pp. 153-201
    • LI, P.1
  • 41
    • 84980078895 scopus 로고
    • A Hamack inequality for parabolic differential equations
    • [Mo]
    • [Mo] J. MOSER, A Hamack inequality for parabolic differential equations. Comm. Pure Appi. Math. 17 (1964), 101-134.
    • (1964) Comm. Pure Appi. Math , vol.17 , pp. 101-134
    • MOSER, J.1
  • 42
    • 0001220619 scopus 로고
    • Uniform restricted parabolic Hamack inequality, separation principle, and ultracontractivity for parabolic equations
    • [Ml] H. Komatsu (ed), Springer-Verlag, Berlin
    • [Ml] M. MURATA, Uniform restricted parabolic Hamack 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.
    • (1993) Functional Analysis and Related Topics", 1991, Lecture Notes in Math , vol.1540 , pp. 277-285
    • MURATA, M.1
  • 43
    • 0002164668 scopus 로고
    • Non-uniqueness of the positive Cauchy problem for parabolic equations
    • [M2]
    • [M2] M. MURATA, Non-uniqueness of the positive Cauchy problem for parabolic equations, J. Differential Equations 123 (1995), 343-387.
    • (1995) J. Differential Equations , vol.123 , pp. 343-387
    • MURATA, M.1
  • 44
    • 0040521112 scopus 로고
    • Sufficient condition for non-uniqueness of the positive Cauchy problem for parabolic equations
    • [M3] Kinokuniya, Tokyo
    • [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.
    • (1994) Spectral and Scattering Theory and Applications", Advanced Studies in Pure Math , vol.23 , pp. 275-282
    • MURATA, M.1
  • 45
    • 0001554104 scopus 로고
    • Uniqueness and non-uniqueness of the positive Cauchy problem for the heat equation on Riemannian manifolds
    • [M4]
    • [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.
    • (1995) Proc. Amer. Math. Soc , vol.123 , pp. 1923-1932
    • MURATA, M.1
  • 46
    • 0030080042 scopus 로고    scopus 로고
    • Non-uniqueness of the positive Dirichlet problem for parabolic equations in cylinders
    • [M5]
    • [M5] M. MURATA, Non-uniqueness of the positive Dirichlet problem for parabolic equations in cylinders, J. Fune. Anal. 135 (1996), 456-487.
    • (1996) J. Fune. Anal , vol.135 , pp. 456-487
    • MURATA, M.1
  • 47
    • 0031498748 scopus 로고    scopus 로고
    • Semismall perturbations in the Martin theory for elliptic equations
    • [M6]
    • [M6] M. MURATA, Semismall perturbations in the Martin theory for elliptic equations, Israel J. Matii. 102 (1997), 29-60.
    • (1997) Israel J. Matii , vol.102 , pp. 29-60
    • MURATA, M.1
  • 48
    • 0039335979 scopus 로고
    • Elliptic operators preserving probability
    • [PS]
    • [PS] M. A. PEREL'MUTER - YU. A. SEMENOV, Elliptic operators preserving probability. Theory Prob. Appi. 32 (1987), 718-721.
    • (1987) Theory Prob. Appi , vol.32 , pp. 718-721
    • PEREL'MUTER, M. A.1    SEMENOV, YU. A.2
  • 49
    • 0030364679 scopus 로고    scopus 로고
    • On uniqueness and nonuniqueness of positive Cauchy problem for parabolic equations with unbounded coefficients
    • [Pine]
    • [Pine] Y. PINCHOVER, On uniqueness and nonuniqueness of positive Cauchy problem for parabolic equations with unbounded coefficients. Math. Z. 233 (1996), 569-586.
    • (1996) Math. Z , vol.233 , pp. 569-586
    • PINCHOVER, Y.1
  • 51
    • 84972525827 scopus 로고
    • Uniformly elliptic operators on Riemannian manifolds
    • [Sal]
    • [Sal] L . SALOFF-COSTE, Uniformly elliptic operators on Riemannian manifolds, J . Differential Geom. 36 (1992), 417-450.
    • (1992) J . Differential Geom , vol.36 , pp. 417-450
    • SALOFF-COSTE, L .1
  • 52
    • 84963021360 scopus 로고
    • A note on Poincaré, Sobolev, and Hamack inequality
    • [Sa2]
    • [Sa2] L. SALOFF-COSTE, A note on Poincaré, Sobolev, and Hamack inequality, Duke Math. J . 2 (1992), 27-38.
    • (1992) Duke Math. J , vol.2 , pp. 27-38
    • SALOFF-COSTE, L.1
  • 53
    • 0001586659 scopus 로고
    • Parabolic Hamack inequality for divergence form second order differential operators
    • [Sa3]
    • [Sa3] L . SALOFF-COSTE, Parabolic Hamack inequality for divergence form second order differential operators. Potential Anal. 4 (1995), 429-467.
    • (1995) Potential Anal , vol.4 , pp. 429-467
    • SALOFF-COSTE, L .1
  • 54
    • 0001598648 scopus 로고
    • Analysis on local Dirichlet spaces-I. Recurrence, conservativeness and LP-Liouville properties
    • [Stul]
    • [Stul] K . - T H . STURM, Analysis on local Dirichlet spaces-I. Recurrence, conservativeness and LP-Liouville properties, J. Reine Angew. Math. 456 (1994), 173-196.
    • (1994) J. Reine Angew. Math , vol.456 , pp. 173-196
    • STURM, K . - T H .1
  • 55
    • 84972506714 scopus 로고
    • Analysis on local Dirichlet spaces-IL Upper Gaussian estimates for the fundamental solutions of parabolic equations
    • [Stu2]
    • [Stu2] K . - T H . STURM, Analysis on local Dirichlet spaces-IL Upper Gaussian estimates for the fundamental solutions of parabolic equations, Osaka J. Matíi. 32 (1995), 275-312.
    • (1995) Osaka J. Matíi , vol.32 , pp. 275-312
    • STURM, K . - T H .1
  • 56
    • 0001581975 scopus 로고    scopus 로고
    • Analysis on local Dirichlet spaces-Ill. Poincaré and parabolic Hamack inequality
    • [Stu3]
    • [Stu3] K . - T H . STURM, Analysis on local Dirichlet spaces-Ill. Poincaré and parabolic Hamack inequality, J. Matii. Pures Appi. (9) 75 (1996), 273-297.
    • (1996) J. Matii. Pures Appi , vol.75 , Issue.9 , pp. 273-297
    • STURM, K . - T H .1
  • 57
    • 0001604599 scopus 로고
    • On the geometry defined by Dirichlet forms
    • [Stu4] E. Bolthansen et al. (eds) (Progress in Prob, Birkhäuser
    • [Stu4] K . - T H . STURM, On the geometry defined by Dirichlet forms. In: "Seminar on Stochastic Analysis, Random Fields and Applications", E. Bolthansen et al. (eds.) (Progress in Prob, vol. 36), Birkhäuser, 1995, pp. 231-242.
    • (1995) Seminar on Stochastic Analysis, Random Fields and Applications , vol.36 , pp. 231-242
    • STURM, K . - T H .1
  • 58
    • 0002585081 scopus 로고
    • Sur les classes quasianalytiques des solutions des équations aux dérivées partielles du type parabolique
    • TÄCKLIND, Sur les classes quasianalytiques des solutions des équations aux dérivées partielles du type parabolique. Nova Acta Regiae Soc. Seien. Upsaliensis, Ser. IV 10 (1936), 1-57.
    • (1936) Nova Acta Regiae Soc. Seien. Upsaliensis, Ser. IV , vol.10 , pp. 1-57
    • TÄCKLIND1
  • 59
    • 84966240246 scopus 로고
    • Positive temperatures on an infinite rod
    • D . V . WIDDER, Positive temperatures on an infinite rod. Trans. Amer. Math. Soc. 55 (1944), 85-95.
    • (1944) Trans. Amer. Math. Soc , vol.55 , pp. 85-95
    • WIDDER, D . V .1


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