Backward uniqueness for parabolic operators with non-Lipschitz coefficients. (English) Zbl 1334.35074

In the present paper, the uniqueness property for the backward parabolic operator is investigated. The main result indicates some sufficient conditions assuring the uniqueness property. Involved are moduli of continuity, the Osgood condition and assumptions on the coefficients of the operator. Some basics are introduced first: modulus of continuity, the Osgood condition, results of the Littlewood-Paley theory, and Bony’s paraproduct. Furthermore, starting from a modulus of continuity, a weight function is constructed and a Carleman estimate is proved. The uniqueness result is a consequence of this estimate.


35K15 Initial value problems for second-order parabolic equations
35R25 Ill-posed problems for PDEs
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
Full Text: arXiv Euclid


[1] J.-M. Bony: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires , Ann. Sci. École Norm. Sup. (4) 14 (1981), 209-246.
[2] R.R. Coifman and Y. Meyer: Au Delà des Opérateurs Pseudo-Différentiels, Astérisque 57 , Soc. Math. France, Paris, 1978. · Zbl 0435.30029
[3] F. Colombini and N. Lerner: Hyperbolic operators with non-Lipschitz coefficients , Duke Math. J. 77 (1995), 657-698. · Zbl 0840.35067 · doi:10.1215/S0012-7094-95-07721-7
[4] D. Del Santo: A remark on the uniqueness for backward parabolic operators with non- Lipschitz-continuous coefficients; in Evolution Equations of Hyperbolic and Schrödinger Type, Progr. Math. 301 , Birkhäuser/Springer Basel AG, Basel, 2012, 103-114. · Zbl 1258.35004 · doi:10.1007/978-3-0348-0454-7_6
[5] D. Del Santo and M. Prizzi: Backward uniqueness for parabolic operators whose coefficients are non-Lipschitz continuous in time , J. Math. Pures Appl. (9) 84 (2005), 471-491. · Zbl 1074.35042 · doi:10.1016/j.matpur.2004.09.004
[6] D. Del Santo and M. Prizzi: A new result on backward uniqueness for parabolic operators , Ann. Mat. Pura Appl. (4) 194 (2015), 387-403, DOI 10.1007/s10231-013-0381-3. · Zbl 1320.35005 · doi:10.1007/s10231-013-0381-3
[7] J.-L. Lions and B. Malgrange: Sur l’unicité rétrograde dans les problèmes mixtes paraboliques , Math. Scand. 8 (1960), 277-286. · Zbl 0126.12202 · doi:10.7146/math.scand.a-10611
[8] N. Mandache: On a counterexample concerning unique continuation for elliptic equations in divergence form , Math. Phys. Anal. Geom. 1 (1998), 273-292. · Zbl 0920.35034 · doi:10.1023/A:1009745125885
[9] G. Métivier: Para-Differential Calculus and Applications to the Cauchy Problem for Nonlinear Systems, Centro di Ricerca Matematica Ennio De Giorgi (CRM) Series 5 , Edizioni della Normale, Pisa, 2008.
[10] K. Miller: Nonunique continuation for uniformly parabolic and elliptic equations in self-adjoint divergence form with Hölder continuous coefficients , Arch. Rational Mech. Anal. 54 (1974), 105-117. · Zbl 1094.14052 · doi:10.5565/PUBLMAT_49105_01
[11] P.C. Rosenbloom and D.V. Widder: A temperature function which vanishes initially , Amer. Math. Monthly 65 (1958), 607-609. · Zbl 0083.09003 · doi:10.2307/2309117
[12] S. Tarama: Local uniqueness in the Cauchy problem for second order elliptic equations with non-Lipschitzian coefficients , Publ. Res. Inst. Math. Sci. 33 (1997), 167-188. · Zbl 0882.35034 · doi:10.2977/prims/1195145537
[13] M.E. Taylor: Pseudodifferential Operators and Nonlinear PDE, Progress in Mathematics 100 , Birkhäuser Boston, Boston, MA, 1991. · Zbl 0746.35062
[14] M.E. Taylor: Tools for PDE, Mathematical Surveys and Monographs 81 , Amer. Math. Soc., Providence, RI, 2000. · Zbl 0963.35211
[15] A. Tychonoff: Théorèmes d’unicité pour l’équation de la chaleur , Mat. Sb. 42 (1935), 199-216. · Zbl 0012.35501
[16] C. Zuily: Uniqueness and Nonuniqueness in the Cauchy Problem, Progress in Mathematics 33 , Birkhäuser Boston, Boston, MA, 1983. · Zbl 0792.35065 · doi:10.2307/2374966
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.