Advertisement

On the Nonlinear Cauchy Problem

  • Massimo Cicognani
  • Luisa Zanghirati
Chapter
  • 633 Downloads
Part of the Operator Theory: Advances and Applications book series (OT, volume 159)

Abstract

Our aim is to describe how to obtain, with the same procedure, several results of local existence, uniqueness and propagation of regularity for the solution of a quasilinear hyperbolic Cauchy Problem.

Keywords

Nonlinear hyperbolic Cauchy problem regularity of solutions 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    R. Agliardi and M. Cicognani, The Cauchy problem for a class of Kovalevskian pseudo-differential operators, Proc. Amer. Math. Soc. 132 (2004), 841–845.CrossRefMathSciNetGoogle Scholar
  2. [2]
    S. Alinhac and G. Metivier, Propagation de l’analycité des solutions de systémes hyperboliques non-linéaires, Invent. Math. 75 (1984), 189–203.CrossRefMathSciNetGoogle Scholar
  3. [3]
    A. Ascanelli, Quasilinear hyperbolic operators with log-Lipschitz coefficients, J. Math. Anal. Appl. 295 (2004), 70–79.CrossRefzbMATHMathSciNetGoogle Scholar
  4. [4]
    A. Ascanelli, The Cauchy problem for quasilinear operators with non-absolutely continuous coefficients, Preprint Math. Dep. Ferrara Univ. (2004).Google Scholar
  5. [5]
    M. Beals, Propagation and interaction of singularities in nonlinear hyperbolic problems, Birkhäuser, Boston, 1989.Google Scholar
  6. [6]
    G. Bourdaud, M. Reissig and W. Sickel, Hyperbolic equations, function spaces with exponential weights and Nemytskij operators, Ann. Mat. Pura Appl. 182 (2003), 409–455.CrossRefMathSciNetGoogle Scholar
  7. [7]
    M. D. Bronštein, The Cauchy problem for hyperbolic operators with characteristics of variable multiplicity, Trudy Moskov. Mat. Obshch. 41 (1980), 83–99.MathSciNetGoogle Scholar
  8. [8]
    J. Chazarain, Opérateurs hyperboliques a caractéristiques de multiplicité constante, Ann. Inst. Fourier 24 (1974), 173–202.zbMATHMathSciNetGoogle Scholar
  9. [9]
    J. Chazarain, Propagation des singularités pour une classe d’opérateurs a caractéristiques multiples et résolubilité locale, Ann. Inst. Fourier 24 (1974), 203–223.zbMATHMathSciNetGoogle Scholar
  10. [10]
    M. Cicognani, Weakly hyperbolic equations with Lipschitz or Hölder continuous coefficients with respect to time, Ann. Univ. Ferrara Sez. VII (N.S.) 38 (1992), 193–215.zbMATHMathSciNetGoogle Scholar
  11. [11]
    M. Cicognani, On the strictly hyperbolic equations which are Hölder continuous with respect to time, Italian J. Pure Appl. Math. 4 (1998), 73–82.zbMATHMathSciNetGoogle Scholar
  12. [12]
    M. Cicognani, The Cauchy problem for strictly hyperbolic operators with nonabsolutely continuous coefficients, Tsukuba J. Math. 27 (2003), 1–12.zbMATHMathSciNetGoogle Scholar
  13. [13]
    M. Cicognani, Coefficients with unbounded derivatives in hyperbolic equations, Math. Nachr. 276, (2004), 31–46.CrossRefzbMATHMathSciNetGoogle Scholar
  14. [14]
    M. Cicognani and L. Zanghirati, The Cauchy problem for nonlinear equations with Levi conditions, Bull. Sci. Math. 123 (1999), 413–435.MathSciNetGoogle Scholar
  15. [15]
    M. Cicognani and L. Zanghirati, Nonlinear weakly hyperbolic equations with Levi conditions in Gevrey classes, Tsukuba J. Math. 25 (2001), 85–102.MathSciNetGoogle Scholar
  16. [16]
    M. Cicognani and L. Zanghirati, Nonlinear hyperbolic Cauchy problems in Gevrey classes, Chinese Ann. Math. Ser. B 22 (2001), 417–426.MathSciNetGoogle Scholar
  17. [17]
    M. Cicognani and L. Zanghirati, Analytic regularity for solutions of nonlinear weakly hyperbolic equations, Boll. Un. Mat. Ital. (B) 11 (1997), 643–679.MathSciNetGoogle Scholar
  18. [18]
    M. Cicognani and L. Zanghirati, Analytic regularity for solutions to semilinear weakly hyperbolic equations, Rend. Sem. Mat. Univ. Politec. Torino 51 (1993), 387–396.MathSciNetGoogle Scholar
  19. [19]
    M. Cicognani and L. Zanghirati, Propagation of Gevrey and analytic regularity for a class of semilinear weakly hyperbolic equations, Rend. Sem. Mat. Univ. Padova 94 (1995), 99–111.MathSciNetGoogle Scholar
  20. [20]
    F. Colombini, E. De Giorgi and S. Spagnolo, Sur les équations hyperboliques avec des coefficients qui ne dépendent que du temps. Ann. Sc. Norm. Sup. Pisa Cl. Sci. 6 (1979), 511–559.Google Scholar
  21. [21]
    F. Colombini, D. Del Santo and T. Kinoshita, Well-posedness of the Cauchy problem for a hyperbolic equation with non-Lipschitz coefficients, Ann. Sc. Norm. Sup. Pisa Cl. Sci. 1 (2002), 327–358.Google Scholar
  22. [22]
    F. Colombini, D. Del Santo and M. Reissig, On the optimal regularity coefficients in hyperbolic Cauchy problem, Bull. Sci. Math. 127 (2003), 328–347.MathSciNetGoogle Scholar
  23. [23]
    F. Colombini, E. Jannelli and S. Spagnolo, Well-posedness in the Gevrey classes of the Cauchy problem for a nonstrictly hyperbolic equation with coefficients depending on time, Ann. Sc. Norm. Sup. Pisa Cl. Sci. 10 (1983), 291–312.MathSciNetGoogle Scholar
  24. [24]
    F. Colombini and N. Lerner, Hyperbolic operators with non-Lipschitz coefficients, Duke Math. J. 77 (1995), 657–698.CrossRefMathSciNetGoogle Scholar
  25. [25]
    P. D’Ancona and S. Spagnolo, Quasi-symmetrization of hyperbolic systems and propagation of analytic regularity, Boll. Un. Mat. Ital. Sez. B Artic. Ric. Mat. 1 (1998), 169–185.MathSciNetGoogle Scholar
  26. [26]
    J.C. De Paris, Problème de Cauchy oscillatoire pour un opérateur différentiel à caractéristiques multiples; lien avec l’hyperbolicité, J. Math. Pures Appl. 51 (1972), 231–256.zbMATHMathSciNetGoogle Scholar
  27. [27]
    H. Flascka and G. Strang, The correctness of the Cauchy problem, Adv. Math. 6 (1971), 349–379.Google Scholar
  28. [28]
    D. Gourdin, Une classe d’operateurs faiblement hyperboliques non linéaires, Bull. Sci. Math. 113 (1989), 349–379.MathSciNetGoogle Scholar
  29. [29]
    F. Hirosawa and M. Reissig, Well-posedness in Sobolev spaces for second order strictly hyperbolic equations with non-differentiable oscillating coefficients, Ann. Global Anal. Geom. 25 (2004), 99–119.CrossRefMathSciNetGoogle Scholar
  30. [30]
    V. Ya. Ivrii, Conditions for correctness in Gevrey classes of the Cauchy problem for weakly hyperbolic equations, Siberian Math. J. 17 (1976), 422–435.Google Scholar
  31. [31]
    K. Kajitani, Cauchy problem for nonstrictly hyperbolic systems in Gevrey classes, J. Math. Kyoto Univ. 23 (1983), 599–616.zbMATHMathSciNetGoogle Scholar
  32. [32]
    K. Kajitani, Local solution of Cauchy problem for nonlinear hyperbolic systems in Gevrey spaces, Hokkaido Math. J. 12 (1983), 436–460.MathSciNetGoogle Scholar
  33. [33]
    K. Kajitani, The Cauchy problem for nonlinear hyperbolic systems, Bull. Sci. Math. 110 (1986), 3–48.zbMATHMathSciNetGoogle Scholar
  34. [34]
    H. Komatsu, Linear hyperbolic equations with Gevrey coefficients, J. Math. Pures Appl. 59 (1980), 145–185.zbMATHMathSciNetGoogle Scholar
  35. [35]
    A. Kubo and M. Reissig, C-well-posedness of the Cauchy problem for quasilinear hyperbolic equations with coefficients non-Lipschitz in t and smooth in x, Banach Center Publ. 60 (2003), 131–150.MathSciNetGoogle Scholar
  36. [36]
    A. Kubo and M. Reissig, Construction of parametrix for hyperbolic equations with fast oscillations in non-Lipschitz coefficients, Comm. Partial Differential Equations 28 (2003), 1741–1502.CrossRefMathSciNetGoogle Scholar
  37. [37]
    H. Kumano-Go, Pseudo-differential operators, The MIT Press, Cambridge London, 1982.Google Scholar
  38. [38]
    J. Leray and Y. Ohya, Équations et systèmes non-linéaires, hyperboliques nonstricts, Math. Ann. 170 (1967) 167–205.CrossRefMathSciNetGoogle Scholar
  39. [39]
    E. E. Levi, Caratteristiche multiple e problema di Cauchy, Annali di Mat. 16 (1909), 161–201.zbMATHGoogle Scholar
  40. [40]
    S. Matsuura, On non-strict hyperbolicity, Proc. Internat. Conf. on Funct. Anal. and Related Topics, Univ. of Tokyo Press, Tokyo, 1970, 171–176.Google Scholar
  41. [41]
    S. Mizohata, On the Cauchy problem, Notes and Reports in Mathematics in Science and Engineering, 3, Academic Press, Inc., Orlando, FL; Science Press, Beijing, 1985.Google Scholar
  42. [42]
    S. Mizohata and Y. Ohya, Sur la condition de E.E. Levi concernant des équations hyperboliques, Publ. Res. Inst. Math. Sci. 4 (1968), 511–526.MathSciNetGoogle Scholar
  43. [43]
    A. Montanari, Equivalent forms of Levi Condition, Ann. Univ. Ferrara, Sez. VII (N.S.) 45 (1999), 191–203.zbMATHMathSciNetGoogle Scholar
  44. [44]
    T. Nishitani, Sur les équations hyperboliques à coefficients höldériens en t et de classe de Gevrey en x, Bull. Sci. Math. 107 (1983), 113–138.zbMATHMathSciNetGoogle Scholar
  45. [45]
    Y. Ohya and S. Tarama, Le problème de Cauchy à caractéristiques multiples dans la classe de Gevrey I. Coefficients hölderiens en t, Hyperbolic equations and related topics (Katata/Kyoto, 1984), 273–306, Academic Press, Boston, MA, 1986.Google Scholar
  46. [46]
    M. Reissig, Weakly hyperbolic equations with time degeneracy in Sobolev spaces, Abstr. Appl. Anal. 2 (1997), 239–256.zbMATHMathSciNetGoogle Scholar
  47. [47]
    S. Spagnolo, Some results of analytic regularity for the semi-linear weakly hyperbolic equations of the second order, Nonlinear hyperbolic equations in applied sciences, Rend. Sem. Mat. Univ. Politec. Torino 1988, Special Issue, 203–229.Google Scholar
  48. [48]
    M. E. Taylor, Pseudodifferential operators, Princeton Mathematical Series, 34, Princeton University Press, Princeton, N.J., 1981.Google Scholar
  49. [49]
    J. Vaillant, Données de Cauchy portées par une caractéristique double, dans le cas d’un système linéaire d’équations aux dérivées partielles, rôle des bicaractéristiques, J. Math. Pures Appl. 47 (1968), 1–40.zbMATHMathSciNetGoogle Scholar
  50. [50]
    S. Wakabayashi, The Lax-Mizohata theorem for nonlinear Cauchy problems, Comm. Partial Differential Equations 26 (2001), 1367–1384.CrossRefzbMATHMathSciNetGoogle Scholar
  51. [51]
    K. Yagdjian, The Lax-Mizohata theorem for nonlinear gauge invariant equations, Proceedings of the Second ISAAC Congress, Vol. 2 (Fukuoka, 1999), 1547–1561, Int. Soc. Anal. Appl. Comput., 8, Kluwer Acad. Publ., Dordrecht, 2000.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2005

Authors and Affiliations

  • Massimo Cicognani
    • 1
    • 2
  • Luisa Zanghirati
    • 3
  1. 1.Dipartimento di MatematicaBolognaItaly
  2. 2.Facoltà di Ingegneria IICesenaItaly
  3. 3.Dipartimento di MatematicaFerraraItaly

Personalised recommendations