Abstract
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≤s 1≤2−1/s 0. If the equation is strictly hyperbolic, we may replace the previous condition by 1≤s 1≤s 0.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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. B6 (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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kinoshita, T., Taglialatela, G. Time regularity of the solutions to second order hyperbolic equations. Ark Mat 49, 109–127 (2011). https://doi.org/10.1007/s11512-009-0120-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11512-009-0120-6