Arkiv för Matematik

  • Ark. Mat.
  • Volume 49, Number 1 (2011), 109-127.

Time regularity of the solutions to second order hyperbolic equations

Tamotu Kinoshita and Giovanni Taglialatela

Full-text: Open access


We consider the Cauchy problem for a second order weakly hyperbolic equation, with coefficients depending only on the time variable. We prove that if the coefficients of the equation belong to the Gevrey class $\gamma^{s_{0}}$ and the Cauchy data belong to $\gamma^{s_{1}}$, then the Cauchy problem has a solution in $\gamma^{s_{0}}([0,T^{*}];\gamma^{s_{1}}(\mathbb{R}))$ for some T*>0, provided 1≤s1≤2−1/s0. If the equation is strictly hyperbolic, we may replace the previous condition by 1≤s1s0.

Article information

Ark. Mat., Volume 49, Number 1 (2011), 109-127.

Received: 6 May 2009
First available in Project Euclid: 31 January 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

2010 © Institut Mittag-Leffler


Kinoshita, Tamotu; Taglialatela, Giovanni. Time regularity of the solutions to second order hyperbolic equations. Ark. Mat. 49 (2011), no. 1, 109--127. doi:10.1007/s11512-009-0120-6.

Export citation


  • Colombini, F., De Giorgi, E. and Spagnolo, S., Sur les équations hyperboliques avec des coefficients qui ne dépendent que du temps, Ann. Sc. Norm. Super. Pisa Cl. Sci. 6 (1979), 511–559.
  • Colombini, F., Jannelli, E. and Spagnolo, S., Well posedness in the Gevrey classes of the Cauchy problem for a non strictly hyperbolic equation with coefficients depending on time, Ann. Sc. Norm. Super. Pisa Cl. Sci. 10 (1983), 291–312.
  • Colombini, F. and Nishitani, T., On second order weakly hyperbolic equations and the Gevrey classes, in Workshop on Blow-up and Global Existence of Solutions for Parabolic and Hyperbolic Problems (Trieste, 1999 ), Rend. Ist. Mat. Univ. Trieste 31, suppl. 2, pp. 31–50, Università degli Studi di Trieste, Trieste, 2000.
  • D’Ancona, P., Gevrey well-posedness of an abstract Cauchy problem of weakly hyperbolic type, Publ. Res. Inst. Math. Sci. 24 (1988), 433–449.
  • D’Ancona, P. and Spagnolo, S., Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math. 108 (1992), 247–262.
  • Kinoshita, T., On the wellposedness in the ultradifferentiable classes of the Cauchy problem for a weakly hyperbolic equation of second order, Tsukuba J. Math. 22 (1998), 241–271.
  • Komatsu, H., An analogue of the Cauchy–Kowalevsky theorem for ultradifferentiable functions and a division theorem for ultradistributions as its dual, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 26 (1979), 239–254.
  • Petzsche, H. J., On E. Borel’s theorem, Math. Ann. 282 (1988), 299–313.
  • Rodino, L., Linear Partial Differential Operators in Gevrey Spaces, World Scientific, River Edge, NJ, 1993.
  • Taglialatela, G., An extension of a theorem of Nirenberg and applications to semilinear weakly hyperbolic equations, Boll. Un. Mat. Ital. B 6 (1992), 467–486.
  • Tahara, H., Singular hyperbolic systems. VIII. On the well-posedness in Gevrey classes for Fuchsian hyperbolic equations, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 39 (1992), 555–582.