Strictly hyperbolic equations with coefficients low-regular in time and smooth in space

  • Massimo Cicognani
  • Daniel LorenzEmail author


We consider the Cauchy problem for strictly hyperbolic m-th order partial differential equations with coefficients low-regular in time and smooth in space. It is well-known that the problem is \(L^2\) well-posed in the case of Lipschitz continuous coefficients in time, \(H^s\) well-posed in the case of Log-Lipschitz continuous coefficients in time (with an, in general, finite loss of derivatives) and Gevrey well-posed in the case of Hölder continuous coefficients in time (with an, in general, infinite loss of derivatives). Here, we use moduli of continuity to describe the regularity of the coefficients with respect to time, weight sequences for the characterization of their regularity with respect to space and weight functions to define the solution spaces. We establish sufficient conditions for the well-posedness of the Cauchy problem, that link the modulus of continuity and the weight sequence of the coefficients to the weight function of the solution space. The well-known results for Lipschitz, Log-Lipschitz and Hölder coefficients are recovered.


Higher order strictly hyperbolic Cauchy problem Modulus of continuity Loss of derivatives Pseudodifferential operators 

Mathematics Subject Classification

35S05 35L30 47G30 


  1. 1.
    Agliardi, R., Cicognani, M.: Operators of p-evolution with nonregular coefficients in the time variable. J. Differ. Equ. 202(1), 841–845 (2004). doi: 10.1090/S0002-9939-03-07092-8 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Agliardi, R., Cicognani, M.: The Cauchy problem for a class of Kovalevskian pseudo-differential operators. Proc. Am. Math. Soc. 132(3), 143–157 (2004). doi: 10.1016/j.jde.2004.03.028 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bonet, J., Meise, R., Melikhov, S.M.: A comparison of two different ways to define classes of ultradifferentiable functions. Bull. Belg. Math. Soc. 14(3), 425–444 (2007)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Cicognani, M.: Strictly hyperbolic equations with non regular coefficients with respect to time. Ann. Univ. Ferrara Sez. VII (N.S.) 45, 45–58 (1999)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Cicognani, M., Colombini, F.: Modulus of continuity of the coefficients and loss of derivatives in the strictly hyperbolic Cauchy problem. J. Differ. Equ. 221(1), 143–157 (2006). doi: 10.1016/j.jde.2005.06.019 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Colombini, F., De Giorgi, E., Spagnolo, S.: Sur les équations hyperboliques avec des coefficients qui ne dépendent que du temps. Ann. Scuola. Norm. Sci. 6(3), 511–559 (1979)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Colombini, F., Lerner, N.: Hyperbolic operators with non-Lipschitz coefficients. Duke Math. J. 77(3), 657–698 (1995). doi: 10.1215/S0012-7094-95-07721-7 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hörmander, L.: Linear Partial Differential Operators. Springer, Berlin (1963)CrossRefzbMATHGoogle Scholar
  9. 9.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators. Reprint of the 1994 ed., Springer, Berlin (2007)Google Scholar
  10. 10.
    Jannelli, E.: Regularly hyperbolic systems and Gevrey classes. Ann. Math. Pure Appl. 140(1), 133–145 (1985). doi: 10.1007/BF01776846 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kajitani, K.: Cauchy problem for nonstrictly hyperbolic systems in Gevrey classes. J. Math. Kyoto Univ. 23(3), 599–616 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kajitani, K., Wakabayashi, S.: Microhyperbolic operators in Gevrey classes. Publ. Res. Inst. Math. Sci. 25(2), 169–221 (1989). doi: 10.2977/prims/1195173608 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kajitani, K., Yuzawa, Y.: The Cauchy problem for hyperbolic systems with Hölder continuous coefficients with respect to the time variable. Ann. Scuola. Norm. Sci. 5(4), 465–482 (2006)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Komatsu, H.: Ultradistributions and hyperbolicity. In: Garnir, H.G. (ed.) Boundary Value Problems for Linear Evolution PartialDifferential Equations: Proceedings of the NATO Advanced StudyInstitute held in Liège, Belgium, September 6–17, 1976, pp. 157–173. Springer Netherlands, Dordrecht (1977)Google Scholar
  15. 15.
    Mizohata, S.: The Theory of Partial Differential Equations. Cambridge University Press, New York (1973)zbMATHGoogle Scholar
  16. 16.
    Nishitani, T.: Sur les équations hyperboliques à coefficients höldériens en t et de classe de Gevrey en x. Bull. Sci. Math. 107(2), 113–138 (1983)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Pascu, M.: On the definition of Gelfand-Shilov spaces. Ann. Univ. Buchar. Math. Ser. 1(1), 125–133 (2010)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Pilipović, S., Teofanov, N., Tomić, F.: On a class of ultradifferentiable functions. Novi Sad J. Math. 45(1), 125–142 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Pilipović, S., Teofanov, N., Tomić, F.: Beyond Gevrey regularity. J. Pseudo Differ. Oper. Appl. 7(1), 113–140 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Reich, M.: Superposition in Modulation Spaces with Ultradifferentiable Weights. arXiv:1603.08723 [math.FA] (2016). Accessed 28 Nov 2016

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversita di BolognaBolognaItaly
  2. 2.Institute of Applied Analysis, Faculty of Mathematics and Computer ScienceTU Bergakademie FreibergFreibergGermany

Personalised recommendations