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Transactions of the American Mathematical Society
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
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1978-0477456-4
PII: S 0002-9947(1978)0477456-4
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