Abstract.
We study non-uniqueness of nonnegative solutions of the Cauchy or Dirichlet problem for parabolic equations on domains which are not relatively compact in Riemannian manifolds. By introducing the notion of heat escape, we give a general and sharp sufficient condition for the non-uniqueness: If there is a heat escape, then the Cauchy problem (or Dirichlet problem) has a positive solution with zero initial value (or zero initial-boundary value). We also show, under a general and sharp condition, uniqueness of nonnegative solutions to the Dirichlet problem for parabolic equations.
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Armitage, D.H., Gardiner, S.J.: Classical Potential Theory. Springer-Verlag, London, 2001
Agmon, S.: On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds. In: Methods of functional analysis and theory of elliptic equations, D. Greco (ed), Liguori Editore Naples, 1982, pp. 19–52
Aikawa, H.: Norm estimate of Green operator, perturbation of Green function and integrability of superharmonic functions. Math. Ann. 312, 289–318 (1998)
Aikawa, H., Murata, M.: Generalized Cranston-McConnell inequalities and Martin boundaries of unbounded domains. J. Analyse Math. 69, 137–152 (1996)
Ancona, A.: First eigenvalues and comparison of Green's functions for elliptic operators on manifolds or domains. J. Analyse Math. 72, 45–92 (1997)
Ancona, A., Taylor, J.C.: 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-Verlag, New York, 1992, pp. 15–23
Aronson, D.G.: Non-negative solutions of linear parabolic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22, 607–694 (1968)
Aronson, D.G., Besala, P.: Uniqueness of solutions of the Cauchy problem for parabolic equations. J. Math. Anal. Appl. 13, 516–526 (1966)
Aronson, D.G., Besala, P.: Uniqueness of positive solutions of parabolic equations with unbounded coefficients. Colloq. Math. 18, 126–135 (1967)
Aronson, D.G., Serrin, J.: Local behavior of solutions of quasilinear parabolic equations. Arch. Rat. Mech. Anal. 25, 81–122 (1967)
Azencott, R.: Behavior of diffusion semi-groups at infinity. Bull. Soc. Math. France 102, 193–240 (1974)
Davies, E.B.: L 1 properties of second order elliptic operators. Bull. London Math. Soc. 17, 417–436 (1985)
Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge Univ. Press, Cambridge, 1989
Dodziuk, J.: Maximum principle for parabolic inequalities and the heat flow on open manifolds. Indiana Univ. Math. J. 32, 703–716 (1983)
Donnelly, H.: Uniqueness of the positive solutions of the heat equation. Proc. Amer. Math. Soc. 99, 353–356 (1987)
Eidus, D., Kamin, S.: The filtration equation in a class of functions decreasing at infinity. Proc. Amer. Math. Soc. 120, 825–830 (1994)
Feller, W.: The parabolic differential equations and the associated semi-groups of transformations. Ann. Math. 55, 468–519 (1952)
Fabes, E.B., Stroock, D.W.: A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash. Arch. Rat. Mech. Anal. 96, 327–338 (1986)
Grigor'yan, A., Hansen, W.: A Liouville property for Schrödinger operators. Math. Ann. 312, 659–716 (1998)
Grigor'yan, A.: On stochastically complete manifolds. Soviet Math. Dokl. 34, 310–313 (1987)
Grigor'yan, A.: Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Amer. Math. Soc. 36, 135–249 (1999)
Grigor'yan, A.: Estimates of heat kernels on Riemannian manifolds. In: Spectral theory and geometry, London Math. Soc. Lecture Note Ser. vol. 273, B. Davies et al., (eds). Cambridge Univ. Press, Cambridge, 1999, pp. 140–225
Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin, 1977
Helms, L.L.: Introduction to Potential Theory. Wiley-Interscience, New York, 1969
Il'n, A.M., Kalashnikov, A.S., Oleinik, O.A.: Linear equations of the second order of parabolic type. Russian Math. Surveys 17, 1–144 (1972)
Ishige, K., Murata, M.: An intrinsic metric approach to uniqueness of the positive Cauchy problem for parabolic equations. Math. Z. 227, 313–335 (1998)
Ishige, K., Murata, M.: Uniqueness of nonnegative solutions of the Cauchy problem for parabolic equations on manifolds or domains. Ann. Scuola Norm. Sup. Pisa XXX, 171–223 (2001)
Ishige, K.: On the behavior of the solutions of degenerate parabolic equations. Nagoya Math. J. 155, 1–26 (1999)
Ishige, K.: An intrinsic metric approach to uniqueness of the positive Dirichlet problem for parabolic equations in cylinders. J. Diff. Eq. 158, 251–290 (1999)
Khas'minskii, R.Z.: Ergodic properties of recurrent diffusion processes and stabilization of the solution to the Cauchy problem for parabolic equations. Theory of Prob. Appl. 5, 179–196 (1960)
Koranyi, A., Taylor, J.C.: Minimal solutions of the heat equations and uniqueness of the positive Cauchy problem on homogeneous spaces. Proc. Amer. Math. Soc. 94, 273–278 (1985)
Lions, J.L., Magenes, E.: Non-homogeneous boundary value problems and applications. vol. I, Springer-Verlag, Berlin-Heidelberg-New York, 1972
Lin, V., Pinchover, Y.: Manifolds with group actions and elliptic operators. Memoirs Amer. Math. Soc. 112 540 (1994)
Li, P., Yau, S.T.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156, 153–201 (1986)
Moser, J.: A Harnack inequality for parabolic differential equations. Comm. Pure Appl. Math. 17, 101–134 (1964)
Murata, M.: Structure of positive solutions to (-△+V)u=0 in ℝn. Duke Math. J 53, 869–943 (1986)
Murata, M.: Uniform restricted parabolic Harnack inequality, separation principle, and ultra- contractivity 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
Murata, M.: Non-uniqueness of the positive Cauchy problem for parabolic equations. J. Diff. Eq. 123, 343–387 (1995)
Murata, M.: 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
Murata, M.: Uniqueness and non-uniqueness of the positive Cauchy problem for the heat equation on Riemannian manifolds. Proc. Amer. Math. Soc. 123, 1923–1932 (1995)
Murata, M.: Non-uniqueness of the positive Dirichlet problem for parabolic equations in cylinders. J. Func. Anal. 135, 456–487 (1996)
Murata, M.: Semismall perturbations in the Martin theory for elliptic equations. Israel J. Math. 102, 29–60 (1997)
Murata, M.: Martin boundaries of elliptic skew products, semismall perturbations, and fundamental solutions of parabolic equations. J. Funct. Anal. 194, 53–141 (2002)
Pinchover, Y.: On uniqueness and nonuniqueness of positive Cauchy problem for parabolic equations with unbounded coefficients. Math. Z. 233, 569–586 (1996)
Pinsky, R.G.: Positive Harmonic Functions and Diffusion. Cambridge Univ. Press, Cambridge, 1995
Perel'muter, M.A., Semenov, Y.U.A.: Elliptic operators preserving probability. Theory of Probability and Its Applications 32, 718–721 (1987)
Saloff-Coste, L.: Uniformly elliptic operators on Riemannian manifolds. J. Diff. Geom. 36, 417–450 (1992)
Saloff-Coste, L.: A note on Poincaré, Sobolev, and Harnack inequality. Duke Math. J., I. M. R. N. 2, 27–38 (1992)
Saloff-Coste, L.: Parabolic Harnack inequality for divergence form second order differential operators. Potential Analysis 4, 429–467 (1995)
Stampacchia, G.: Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15, 189–258 (1965)
Sturm, K.-Th.: Analysis on local Dirichlet spaces-I. Recurrence, conservativeness and L p-Liouville properties. J. Reine Angew. Math. 456, 173–196 (1994)
Sturm, K.-Th.: Analysis on local Dirichlet spaces-II. Upper Gaussian estimates for the fundamental solutions of parabolic equations. Osaka J. Math. 32, 275–312 (1995)
Sturm, K.-Th.: Analysis on local Dirichlet spaces-III. Poincaré and parabolic Harnack inequality. J. Math. Pures Appl. IX. Ser. 75, 273–297 (1996)
Täcklind: Sur les classes quasianalytiques des solutions des équations aux dérivées partielles du type parabolique. Nova Acta Regiae Soc. Scien. Upsaliensis, Ser. IX 10, 1–57 (1936)
Widder, D.V.: Positive temperatures on an infinite rod. Trans. Amer. Math. Soc. 55, 85–95 (1944)
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Mathematics Subject Classification (1991): 31C12, 35K20, 35J25, 35K15, 58G11, 58G03, 31C05
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Murata, M. Heat escape. Math. Ann. 327, 203–226 (2003). https://doi.org/10.1007/s00208-002-0381-x
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DOI: https://doi.org/10.1007/s00208-002-0381-x