Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Uniqueness in the Cauchy problem for parabolic equations

Author: Roger M. Hayne
Journal: Trans. Amer. Math. Soc. 241 (1978), 373-399
MSC: Primary 35K15
MathSciNet review: 0477456
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In a classical paper S. Täcklind (Nova Acta Soc. Sci. Upsaliensis (4) 10 (1936), 1-57) closed the uniqueness question for the Cauchy problem for the heat equation with a general growth hypothesis which was both necessary and sufficient. Täcklind's proof of the sufficiency involved an ingenious bootstrapping comparison technique employing the maximum principle and a comparison function constructed from the Green's function for a half cylinder. G. N. Zolotarev (Izv. Vysš. Učebn. Zaved. Matematika 2 (1958), 118-135) has extended this result using essentially the same technique to show that Täcklind's uniqueness condition remains sufficient for a general second order parabolic equation provided the coefficients are regular enough to permit the existence and estimation of a Green's function. We have now shown, using a new approach which replaces the construction based upon a Green's function by an appropriate comparison solution of the maximizing equation (C. Pucci, Ann. Mat. Pura Appl. 72 (1966), 141-170), that Täcklind's condition is sufficient without any regularity conditions on the coefficients.

References [Enhancements On Off] (What's this?)

  • [1] T. Carleman, Les fonctions quasianalytiques, Coll. Borel, Paris, 1926.
  • [2] E. Holmgren, Sur les solutions quasianalytiques de l'équation de la chaleur, Ark. Mat. 18, (9) (1924), 1-9.
  • [3] M. Krzyański, Sur les solutions des équations du type parabolique determinées dans une région illimitée, Bull. Amer. Math. Soc. 47 (1941), 911-915. MR 0006008 (3:246e)
  • [4] C. Pucci, Operatori ellittici estremanti, Ann. Mat. Pura Appl. 72 (1966), 141-170. MR 0208150 (34:7960)
  • [5] P. C. Rosenbloom and D. V. Widder, A temperature function that vanishes initially, Amer. Math. Monthly 65 (1958), 607-609. MR 0100717 (20:7146)
  • [6] S. Täcklind, Sur les classes quasianalytiques des solutions des équations aux dérivées partielles du type parabolique, Nova Acta Soc. Sci. Upsal. (4) 10 (1936), 1-57.
  • [7] A. N. Tychonoff, Uniqueness theorems for the heat equation, Mat. Sb. 42 (1935), 199-216.
  • [8] G. N. Watson, Treatise on the theory of Bessel functions, Cambridge Univ. Press, London, 1944. MR 0010746 (6:64a)
  • [9] G. N. Zolotarev, The uniqueness of the solution of the Cauchy problem for systems parabolic in the sense of I. G. Petrowski, Izv. Vysš Učebn. Zaved. Matematika 2 (1958), 118-135. (Russian) MR 0153984 (27:3945)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35K15

Retrieve articles in all journals with MSC: 35K15

Additional Information

Keywords: Cauchy problem, uniqueness, extremal operators, second order parabolic equations, partial differential equations, maximum principle, a priori estimate, Bessel functions
Article copyright: © Copyright 1978 American Mathematical Society

American Mathematical Society