The Cauchy problem for hyperbolic systems with Hölder continuous coefficients with respect to the time variable
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 4, pp. 465-482.

We discuss the local existence and uniqueness of solutions of certain nonstrictly hyperbolic systems, with Hölder continuous coefficients with respect to time variable. We reduce the nonstrictly hyperbolic systems to the parabolic ones and by use of the Tanabe-Sobolevski's method and the Banach scale method we construct a semi-group which gives a representation of the solution to the Cauchy problem.

Classification : 35L45,  35A08
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     title = {The {Cauchy} problem for hyperbolic systems with {H\"older} continuous coefficients with respect to the time variable},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
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     publisher = {Scuola Normale Superiore, Pisa},
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Kajitani, Kunihiko; Yuzawa, Yasuo. The Cauchy problem for hyperbolic systems with Hölder continuous coefficients with respect to the time variable. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 4, pp. 465-482. http://www.numdam.org/item/ASNSP_2006_5_5_4_465_0/

[1] M. D. Bronshtein, Cauchy problem for hyperbolic operators with variable multiple characteristics (Russian), Trudy Moscow Math. 41 (1980), 83-99. | MR 611140 | Zbl 0468.35062

[2] F. Colombini, E. Jannelli and S. Spagnolo, Wellposedness in the Gevrey classes of the Cauchy problem for a non strictly hyperbolic equation with coefficients depending on time, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 10 (1983), 291-312. | EuDML 83908 | Numdam | MR 728438 | Zbl 0543.35056

[3] P. D'Ancona, T. Kinoshita and S. Spagnolo, Weakly hyperbolic systems with Hölder continuous coefficients, J. Differential Equations 203 (2004), 64-81. | MR 2070386 | Zbl 1068.35065

[4] K. Kajitani, Local solution of Cauchy problem for nonlinear hyperbolic systems in Gevrey classes, Hokkaido Math. J. 23-3 (1983), 599-616. | MR 721386 | Zbl 0544.35063

[5] P. D'Ancona, T. Kinoshita and S. Spagnolo, Cauchy problem for nonstrictly hyperbolic systems in Gevrey classes, J. Math. Kyoto. Univ. 12 (1983), 434-460. | Zbl 0544.35063

[6] P. D'Ancona, T. Kinoshita and S. Spagnolo, The Cauchy Problem for Nonlinear Hyperbolic Systems, Bull. Sci. Math. 110 (1986), 3-48. | Zbl 0657.35082

[7] P. D'Ancona, T. Kinoshita and S. Spagnolo, Wellposedness in Gevrey class of the Cauchy problem for hyperbolic operators, Bull. Sc. Math. 111 (1987), 415-438. | Zbl 0653.35051

[8] T. Nishitani, Sur les équations hyperboliques à coefficients hölderients en t et de classes de Gevrey en x, Bull. Sci. Math. 107 (1983), 113-138. | MR 704720 | Zbl 0536.35042

[9] Y. Ohya and S. Tarama, Le Problème de Cauchy à caractéristiques multiples dans la classe de Gevrey - coefficients hölderiens en t -, In: “Hyperbolic Equations and Related Topics”, Proc. Taniguchi Internat. Symp. (1984), 273-302. | Zbl 0665.35045

[10] H. Tanabe, “Equations of Evolution”, translated from the Japanese by N. Mugibayashi and H. Haneda. Monographs and Studies in Mathematics, Vol. 6, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979. | MR 533824 | Zbl 0417.35003

[11] Y. Yuzawa, The Cauchy problem for hyperbolic systems with Hölder continuous coefficients with respect to time, J. Differential Equations 219 (2005), 363-374. | MR 2183264 | Zbl 1087.35068