Abstract
We use the Littlewood-Paley decomposition technique to obtain a C∞-well-posedness result for a weakly hyperbolic equation with a finite order of degeneration.
Keywords: Littlewood-Paley decomposition, Hyperbolic equations, C∞-well-posedness, Approximate energy method
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1. Ascanelli, A., Cicognani, M.: Energy estimate and fundamental solution for degenerate hyperbolic Cauchy problems. To appear in J. Differential Equations (2006)
2. Bony, J.M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Sci. École Norm. Sup. (4) 14, no. 2, 209–246 (1981)
3. Colombini, F., De Giorgi, E., Spagnolo, S.: Sur les équations hyperboliques avec des coefficients qui ne dépendent que du temps. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 6, no. 3, 511–559 (1979)
4. Colombini, F., Del Santo, D.: Strictly hyperbolic operators and approximate energies. Analysis and applications–ISAAC 2001 (Berlin), 253–277, Int. Soc. Anal. Appl. Comput. 10, Kluwer Acad. Publ., Dordrecht (2003)
5. Colombini, F., Di Flaviano, M., Nishitani, T.: On the Cauchy problem for a weakly hyperbolic operator: an intermediate case between effective hyperbolicity and Levi condition. Partial differential equations and mathematical physics, Tokyo, (2001), 73–83, Progr. Nonlinear Differential Equations Appl. 52, Birkhäuser, Boston (2003)
6. Colombini, F., Ishida, H., Orrù, N.: On the Cauchy problem for finitely degenerate hyperbolic equations of second order. Ark. Mat. 38, no. 2, 223–230 (2000)
7. Colombini, F., Jannelli, E., Spagnolo, S.: Well-posedness in Gevrey classes of the Cauchy problem for a non-strictly hyperbolic equation with coefficients depending on time. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 10, no. 2, 291–312 (1983)
8. Colombini, F., Lerner, N.: Hyperbolic operators with non-Lipschitz coefficients. Duke Math. J. 77, no. 3, 657–698 (1995)
9. Colombini, F., Nishitani, T.: On finitely degenerate hyperbolic operators of second order. Osaka J. Math. 41, no. 4, 933–947 (2004)
10. Del Santo, D., Kinoshita, T., Reissig, M.: Energy estimates for strictly hyperbolic equations with low regularity in coefficients. To appear (2006)
11. Nishitani, T.: The Cauchy problem for weakly hyperbolic equations of second order. Comm. Partial Differential Equations 5, no. 12, 1273–1296 (1980)
12. Nishitani, T.: The effectively hyperbolic Cauchy problem. The hyperbolic Cauchy problem (by K. Kajitani and T. Nishitani), 71–167, Lecture Notes in Mathematics 1505, Springer-Verlag, Berlin (1991)
13. Ivrii, V.Ja.: Cauchy problem conditions for hyperbolic operators with characteristics of variable multiplicity for Gevrey classes. (Russian). Sibirsk. Mat. Zh. 17, 1256–1270 (1976). English transl.: Siberian Math. 17, 921–931 (1976)
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Cicognani, M., Santo, D.D. & Reissig, M. A dyadic decomposition approach to a finitely degenerate hyperbolic problem. Ann. Univ. Ferrara 52, 281–289 (2006). https://doi.org/10.1007/s11565-006-0021-6
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DOI: https://doi.org/10.1007/s11565-006-0021-6