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Non-uniqueness and Uniqueness in the Cauchy Problem of Elliptic and Backward-Parabolic Equations

Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 44)

Abstract

In this paper we consider the non-uniqueness and the uniqueness property for the solutions to the Cauchy problem for the operators
$$\mathcal{E} u = \partial_t^2 u + \sum _{k,l=1}^n \partial_{x_k} \bigl(a_{kl}(t,x)\partial_{x_l}u\bigr) + \beta(t,x) \partial_t u + \sum_{m=1}^n b_m(t,x)\partial_{x_m}u + c(t,x)u $$
and
$$\mathcal{P} u = \partial_t u + \sum_{k,l=1}^n \partial_{x_k}\bigl(a_{kl}(t,x)\partial_{x_l}u\bigr) + \sum_{m=1}^n b_m(t,x) \partial_{x_m}u + c(t,x)u, $$
where \(\sum_{k,l=1}^{n} a_{kl}(t,x)\xi_{k} \xi_{l} |\xi|^{-2}\geq a_{0} >0\). We study non-uniqueness and uniqueness in dependence of global and local regularity properties of the coefficients of the principal part. The global regularity will be ruled by the modulus of continuity of a kl on [0,T] while the local regularity will concern a bound on | t a kl (t,x)| on every interval [ε,T]⊆(0,T]. By suitable counterexamples we show that our conditions seem to be sharp in many cases and we compare our statements with known results in the theory of hyperbolic Cauchy problems. We make also some remarks on continuous dependence for \(\mathcal{P}\).

Keywords

Uniqueness Non-uniqueness Cauchy problem Elliptic equations Backward Cauchy problem Parabolic equations Modulus of continuity Osgood condition 

Mathematics Subject Classification

35Bxx 35J15 35K10 35B35 35B60 

Notes

Acknowledgements

Since most of the results are content of his Master thesis the second author would like to thank his supervisor Professor Michael Reissig for turning his attention to the subject of this paper and many helpful and clarifying discussions. He also thanks the Department of Mathematics and Geosciences of the University of Trieste for its warm and inspiring hospitality during several stays in Trieste.

References

  1. 1.
    Agmon, S., Nirenberg, L.: Lower bounds and uniqueness theorems for solutions of differential equations in a Hilbert space. Commun. Pure Appl. Math. 20, 207–229 (1967) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bardos, C., Tatar, L.: Sur l’unicité rétrograde des équations paraboliques et quelques questions voisines. Arch. Ration. Mech. Anal. 50(1), 10–25 (1973). doi: 10.1007/BF00251291 MATHCrossRefGoogle Scholar
  3. 3.
    Bers, L., Nirenberg, L.: On a representation theorem for linear elliptic systems with discontinuous coefficients and its applications. In: Convegno Internazionale sulle Equazioni Lineari alle Derivate Parziali, Trieste, 1954, pp. 141–167. Edizioni Cremonese, Roma (1955) Google Scholar
  4. 4.
    Carleman, T.: Sur un probléme d’unicité pour les systèmes d’équations aux dérivées partielles à deux variables indépendantes. Ark. Mat. Astron. Fys. 26(7), 1–9 (1939) MathSciNetMATHGoogle Scholar
  5. 5.
    Cicognani, M., Hirosawa, F., Reissig, M.: The Log-effect for p-evolution type models. J. Math. Soc. Jpn. 60(3), 819–863 (2008) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Colombini, F., Del Santo, D.: Some uniqueness results for degenerate elliptic operators in two variables. Rend. Semin. Mat. Univ. Padova 86, 111–129 (1991) MathSciNetMATHGoogle Scholar
  7. 7.
    Colombini, F., Del Santo, D.: On the uniqueness in Gevrey spaces for degenerate elliptic operators. Commun. Partial Differ. Equ. 19(11–12), 1945–1969 (1994) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Colombini, F., Del Santo, D.: An example of non-uniqueness for a hyperbolic equation with non-Lipschitz-continuous coefficients. J. Math. Kyoto Univ. 42(3), 517–530 (2002) MathSciNetMATHGoogle Scholar
  9. 9.
    Colombini, F., Lerner, N.: Hyperbolic operators having non-Lipschitz coefficients. Duke Math. J. 77, 657–698 (1995) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Colombini, F., De Giorgi, E., Spagnolo, S.: Sur les équations hyperboliques avec des coefficients qui ne dépendent que du temps. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) 6(3), 511–559 (1979) MATHGoogle Scholar
  11. 11.
    Colombini, F., Del Santo, D., Zuily, C.: Uniqueness and non-uniqueness in the Cauchy problem for a degenerate elliptic operator. Am. J. Math. 115, 1281–1297 (1993) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Colombini, F., Del Santo, D., Kinoshita, T.: Well-posedness of the Cauchy problem for a hyperbolic equation with non-Lipschitz coefficients. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 1(2), 327–358 (2002) MathSciNetMATHGoogle Scholar
  13. 13.
    Colombini, F., Del Santo, D., Reissig, M.: The optimal regularity of coefficients in the hyperbolic Cauchy problem. Bull. Sci. Math. 127, 328–347 (2003) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Del Santo, D.: Nonuniqueness in the Cauchy problem for a degenerate elliptic operator. Bull. Sci. Math. 114(3), 337–350 (1990) MathSciNetMATHGoogle Scholar
  15. 15.
    Del Santo, D.: A remark on non-uniqueness in the Cauchy problem for elliptic operators having non-Lipschitz coefficients. In: Hyperbolic Differential Operators and Related Problems. Lecture Notes in Pure and Applied Mathematics, vol. 233, pp. 317–320 (2003) Google Scholar
  16. 16.
    Del Santo, D.: A remark on the uniqueness for backward parabolic operators with non-Lipschitz-continuous coefficients. In: Evolution Equations of Hyperbolic and Schrödinger Type. Progress in Mathematics, vol. 301, pp. 103–114. Birkhäuser, Basel (2012) CrossRefGoogle Scholar
  17. 17.
    Del Santo, D., Jäh, Ch.P., Paicu, M.: Backward uniqueness for parabolic operators with coefficients with Osgood regularity in time and non-Lipschitz regularity in space. In preparation Google Scholar
  18. 18.
    Del Santo, D., Jäh, Ch.P., Prizzi, M.: Continuous dependence for backward-parabolic operators with coefficients Log-Lipschitz in time and Lipschitz in space. In preparation Google Scholar
  19. 19.
    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). doi: 10.1016/j.matpur.2004.09.004 MathSciNetMATHGoogle Scholar
  20. 20.
    Del Santo, D., Prizzi, M.: Continuous dependence for backward parabolic operators with Log-Lipschitz coefficients. Math. Ann. 345(1), 213–243 (2009). doi: 10.1007/s00208-009-0353-5 MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Del Santo, D., Prizzi, M.: A new result on backward uniqueness for parabolic operators. arXiv:1112.2472 (2012)
  22. 22.
    Del Santo, D., Kinoshita, T., Reissig, M.: Energy estimates for strictly hyperbolic equations with low regularity in coefficients. Differ. Integral Equ. 20(8), 879–900 (2007) MathSciNetMATHGoogle Scholar
  23. 23.
    Ghidaglia, J.-M.: Some backward uniqueness results. Nonlinear Anal., Theory Methods Appl. 10(8), 777–790 (1986). doi: 10.1016/0362-546X(86)90037-4 MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Glagoleva, R.Ja.: Continuous dependence on initial data of the solution to the first boundary-value problem for a parabolic equation with negative time. Sov. Math. Dokl. 4(1), 13–17 (1977) Google Scholar
  25. 25.
    Hirosawa, F., Reissig, M.: About the optimality of oscillations in non-Lipschitz coefficients for strictly hyperbolic equations. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 3, 589–608 (2004) MathSciNetMATHGoogle Scholar
  26. 26.
    Hörmander, L.: On the uniqueness of the Cauchy problem. Math. Scand. 6, 213–225 (1958) MathSciNetMATHGoogle Scholar
  27. 27.
    Hörmander, L.: On the uniqueness of the Cauchy problem II. Math. Scand. 7, 177–190 (1959) MathSciNetMATHGoogle Scholar
  28. 28.
    Hörmander, L.: Linear Partial Differential Operators. Springer, Berlin (1963) MATHCrossRefGoogle Scholar
  29. 29.
    Hurd, A.E.: Backward continuous dependence for mixed parabolic problems. Duke Math. J. 34, 493–500 (1967) MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Jäh, Ch.P.: Uniqueness and non-uniqueness in the Cauchy problem for elliptic and backward parabolic operators. Diplomarbeit, Technische Universität Bergakademie, Freiberg (2010) Google Scholar
  31. 31.
    John, F.: Continuous dependence on data for solutions of partial differential equations with a prescribed bound. Commun. Pure Appl. Math. 13(4), 551–585 (1960). doi: 10.1002/cpa.3160130402 MATHCrossRefGoogle Scholar
  32. 32.
    Kinoshita, T., Reissig, M.: About the loss of derivatives for strictly hyperbolic equations with non-Lipschitz coefficients. Adv. Differ. Equ. 10(2), 191–222 (2005) MathSciNetMATHGoogle Scholar
  33. 33.
    Lerner, N.: Résultats d’unicité forte pour des opérateurs elliptiques à coefficients Gevrey. Commun. Partial Differ. Equ. 6(10), 1163–1177 (1981). doi: 10.1080/03605308108820208 MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Lions, J.-L., Malgrange, B.: Sur l’unicité rétrograde dans les problèmes mixtes paraboliques. Math. Scand. 8, 277–286 (1960) MathSciNetGoogle Scholar
  35. 35.
    Mizohata, S.: Le problème de Cauchy pour le passé pour quelques équations paraboliques. Proc. Jpn. Acad. 34(10), 693–696 (1958) MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Nirenberg, L.: Uniqueness in the Cauchy problem for a degenerate elliptic second order equation. In: Differential Geometry and Complex Analysis, pp. 213–218. Springer, Berlin (1985) CrossRefGoogle Scholar
  37. 37.
    Oleinik, O.A.: On the Cauchy problem for weakly hyperbolic equations. Commun. Pure Appl. Math. 23, 569–586 (1970). doi: 10.1002/cpa.3160230403 MathSciNetCrossRefGoogle Scholar
  38. 38.
    Osgood, W.F.: Beweis der Existenz einer Lösung der Differentialgleichung \(\frac{dy}{dx}=f(x,y)\) ohne Hinzunahme der Cauchy Lipschitz’schen Bedingung. Monatshefte Math. 9, 331–345 (1898) MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Pliš, A.: On non-uniqueness in Cauchy problem for an elliptic second order differential equation. Bull. Acad. Pol. Sci., Sér. Sci. Math. 11, 95–100 (1963) MATHGoogle Scholar
  40. 40.
    Reissig, M., Yagdjian, K.: About the influence of oscillations on Strichartz-type decay estimates. Rend. Semin. Mat. (Torino) 58(3), 375–388 (2000) MathSciNetMATHGoogle Scholar
  41. 41.
    Tarama, S.: Local uniqueness in the Cauchy problem for second order elliptic equations with non-Lipschitzian coefficients. Publ. Res. Inst. Math. Sci. 33(1), 167–188 (1997). doi: 10.2977/prims/1195145537 MathSciNetCrossRefGoogle Scholar
  42. 42.
    Tarama, S.: Energy estimate for wave equations with coefficients in some Besov type class. Electron. J. Differ. Equ. 85, 1–12 (2007) MathSciNetGoogle Scholar
  43. 43.
    Yamazaki, T.: On the \(L^{2}(\mathbb{R}^{n})\) well-posedness of some singular and degenerate partial differential equations of hyperbolic type. Commun. Partial Differ. Equ. 15, 1029–1078 (1990) MATHCrossRefGoogle Scholar
  44. 44.
    Yagdjian, K.: The Cauchy Problem for Hyperbolic Operators: Multiple Characteristics, Micro-local Approach. Akademie Verlag, Berlin (1997) MATHGoogle Scholar
  45. 45.
    Zuily, C.: Uniqueness and Non-uniqueness in the Cauchy Problem. Progress in Mathematics, vol. 33. Birkhäuser, Boston (1983) MATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  1. 1.Dipartimento di Matematica e GeoscienzeUniversità di TriesteTriesteItaly
  2. 2.Institut für Angewandte AnalysisTU Bergakademie FreibergFreibergGermany

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