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Continuous dependence for backward parabolic operators with Log-Lipschitz coefficients

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We prove continuous dependence on Cauchy data for a backward parabolic operator whose coefficients are Log-Lipschitz continuous in time.

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References

  1. Agmon S., Nirenberg L.: Properties of solutions of ordinary differential equations in Banach space. Comm. Pure Appl. Math. 16, 121–239 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bony J.-M.: Calcul symbolique et propagations des singularités pour les équations aux dérivées partielles non linéaires. Ann. Sci. École Norm. Sup. 14, 209–246 (1981)

    MATH  MathSciNet  Google Scholar 

  3. Colombini F., Lerner N.: Hyperbolic operators having non-Lipschitz coefficients. Duke Math. J. 77, 657–698 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  4. Del Santo D., Prizzi M.: Backward uniqueness for parabolic operators whose coefficients are non-Lipschitz continuous in time. J. Math. Pures Appl. 84, 471–491 (2005)

    MATH  MathSciNet  Google Scholar 

  5. Glagoleva R.Ja.: Continuous dependence on initial data of the solution to the first bounded value problem for a parabolic equation with negative time. Soviet Math. 4(1), 13–17 (1963)

    MATH  Google Scholar 

  6. Halmos P.R., Sunder V.S.: Bounded integral operators on L 2 spaces. Springer-Verlag, Berlin- New York (1978)

    MATH  Google Scholar 

  7. Hurd A.E.: Backward continuous dependence for mixed parabolic problems. Duke Math. J. 34, 493–500 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  8. Isakov V.: Inverse problems for partial differential equations. Springer-Verlag, New York (1998)

    MATH  Google Scholar 

  9. John F.: Continuous dependence on data for solutions of partial differential equations with a prescribed bound. Comm. Pure Appl. Math. 13, 551–585 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  10. Lions J.-L., Malgrange B.: Sur l’unicité rétrograde dans les problèmes mixtes paraboliques. Math. Scand. 8, 277–286 (1960)

    MathSciNet  Google Scholar 

  11. Mandache N.: On a counterexample concerning unique continuation for elliptic equations in divergence form. Math. Phys. Anal. Geom. 1, 273–292 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  12. Miller K.: Nonunique continuation for uniformly parabolic and elliptic equations in self–adjoint divergence form with Hölder continuous coefficients. Arch. Rat. Mech. Anal. 54, 105–117 (1973)

    Google Scholar 

  13. Pliś A.: On non-uniqueness in Cauchy problem for an elliptic second order differential equation. Bull. Ac. Pol. Sci. 11, 95–100 (1963)

    MATH  Google Scholar 

  14. Tychonoff A.: Théorème d’unicité pour l’équation de la chaleur. Rec. math. Moscou 42, 199–216 (1935)

    MATH  Google Scholar 

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Correspondence to Daniele Del Santo.

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Del Santo, D., Prizzi, M. Continuous dependence for backward parabolic operators with Log-Lipschitz coefficients. Math. Ann. 345, 213–243 (2009). https://doi.org/10.1007/s00208-009-0353-5

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  • DOI: https://doi.org/10.1007/s00208-009-0353-5

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