Kyoto Journal of Mathematics

A sufficient condition for well-posedness for systems with time-dependent coefficients

Marcello D’Abbicco

Full-text: Open access


We consider linear, smooth, hyperbolic systems with time-dependent coefficients and size N. We give a condition sufficient for the well-posedness of the Cauchy Problem in some Gevrey classes. We present some Levi conditions to improve the Gevrey index of well-posedness for the scalar equation of order N, using the transformation in [DAS] and following the technique introduced in [CT]. By using this result and adding some assumptions on the form of the first-order term, we can improve the well-posedness for systems. A similar condition has been studied in [DAT] for systems with size 3.

Article information

Kyoto J. Math., Volume 50, Number 2 (2010), 365-401.

First available in Project Euclid: 7 May 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35L45: Initial value problems for first-order hyperbolic systems


D’Abbicco, Marcello. A sufficient condition for well-posedness for systems with time-dependent coefficients. Kyoto J. Math. 50 (2010), no. 2, 365--401. doi:10.1215/0023608X-2009-017.

Export citation


  • [B1] M. D. Bronšteĭn, Smoothness of roots of polynomials depending on parameters, Sibirsk. Mat. Zh. 20 (1979), 493–501; English translation in Siberian Math. J. 20 (1980), 347–352.
  • [B2] M. D. Bronšteĭn, The Cauchy problem for hyperbolic operators with characteristics of variable multiplicity (in Russian), Trudy Moskov. Mat. Obshch. 41 (1980), 83–99.
  • [CI] F. Colombini and H. Ishida, Well-posedness of the Cauchy problem in Gevrey classes for some weakly hyperbolic equations of higher order, J. Anal. Math. 90 (2003), 13–25.
  • [CJS] 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. Scuola. Norm. Sup. Pisa Cl. Sci. (4) 10 (1983), 291–312.
  • [CO] F. Colombini and N. Orrú, Well-posedness in $\mathcalC$ for some weakly hyperbolic equations, J. Math. Kyoto Univ. 39 (1999), 399–420.
  • [CT] F. Colombini and G. Taglialatela, Well-posedness for hyperbolic higher order operators with finite degeneracy, J. Math. Kyoto Univ. 46 (2006), 833–877.
  • [D] M. D’Abbicco, Some results on the Well-Posedness for Second Order Linear Equations, Osaka J. Math. 46 (2009), 739–767.
  • [DAK] P. D’Ancona and T. Kinoshita, On the wellposedness of the Cauchy problem for weakly hyperbolic equations of higher order, Math. Nachr. 278 (2005), 1147–1162.
  • [DAS] P. D’Ancona and S. Spagnolo, Quasi-symmetrization of hyperbolic systems and propagation of the analytic regularity, Boll. Unione Mat. Ital., Sez. B Artic. Ric. Mat. (8) 1 (1998), 169–185.
  • [DAT] M. D’Abbicco and G. Taglialatela, Some results on the well-posedness for systems with time-dependent coefficients, Ann. Fac. Sci. Toulouse Math. (6) 18 (2009), 247–284.
  • [H] L. Hörmander, Linear Partial Differential Operators, Grundlehren Math. Wiss. 116 Springer, Berlin, 1963.
  • [Y] H. Yamahara, Cauchy problem for hyperbolic systems in Gevrey class: A note on Gevrey indices, Ann. Fac. Sci. Toulouse Math. (6) 9 (2000), 147–160.