Abstract
First of all, let us recall the two classical results of references [1] and [2]. These two papers are not related directly to the classical concept of hyperbolicity by J. Hadamard; that is, the well-posed Cauchy problem in the class of infinitely differentiable functions. Therefore, we were obliged to discuss the well-posedness in the class of Gevrey. In other words, we understand hyperbolicity when there is the influence (or dependence) domain extending the traditional hyperbolicity of differential operators.
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References
Y. OhyaLe problème de Cauchy pour les équations hyperboliques à caractéristique multipleJ. Math. Soc. Japan. 16 (1964), 268–286.
J. Leray and Y. OhyaSystèmes linéaires hyperboliques non strictsColloque de Liege CBRM, (1964), 105–144.
P.D. LaxAsymptotic solutions of oscillatory initial value problemsDuke Math. J. 24 (1957), 627–646.
S. MizohataSome remarks on the Cauchy problemJ. Math. Kyoto Univ. 1 (1961), 107–127.
J. HadamardLes fonctions de classe superieure dans l’équation de VolterraJour. d’Anal. Math. 1 (1951).
A. LaxOn Cauchy’s problem for partial differential equations with multiple characteristicsComm. Pure Appl. Math.9(1956).
M. YamagutiLe problème de Cauchy et les opérateurs d’intégrale singulièreMem. Coll. Sci. Kyoto Univ., Series A. 32 (1) (1959).
E.E. LeviCaratteristico multiple e ploblema di CauchyAnn. Math. Pura Appli.16 (1909)109–127.
E. DE GiorgiUn teorema di unicita per it problema di Cauchy relativo ad equatini differenziali lineari a derivate partiali di tipo parabokicoAnnali di Mat.40(1955), 371–377;Un exempio di non unicita della solutione del problema di CauchyUniv. di Roma, Rendiconti di Math.14(1955), 382–387.
L. NirenbergPseudo-differential operators in global analysisProc. Sympo. Pure Math.16(1970), 149–167.
M.D. BronshteinThe Cauchy problem for hyperbolic operators with characteristics of variable multiplicityTrudy Moskov. Mat. Obsc.41(1980), 87103;Smoothness of roots of polynomials depending on parametersSibirsk. Mat. Zeit.20(1970), 493–501.
Y. Ohya et S. TaramaLe problème de Cauchy à caractéristques multiples dans la classe de GevreyTaniguchi Sympo. HEAT, Katata, (1984), 273–306.
S. WakabayashiRemarks on hyperbolic polynomialsTukuba J. Math. 10 (1986), 17–28.
S. TaramaOn the initial value problem for the weakly hyperbolic operators in the Gevrey classesProc. Hyperbolic Equation, Pisa, (1987), 322–339.
W. NuijA note on hyperbolic polynomialsMath. Scand.23(1968), 69–72.
H. Kumano-goPseudo-differential OperatorsM.I.T. Press, Cambridge, 1981.
L. HörmanderLinear Partial Differential OperatorsSpringer-Verlag, 1963.
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Ohya, Y. (2003). Hyperbolic Cauchy Problem Well Posed in the Class of Gevrey. In: Kajitani, K., Vaillant, J. (eds) Partial Differential Equations and Mathematical Physics. Progress in Nonlinear Differential Equations and Their Applications, vol 52. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0011-6_13
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DOI: https://doi.org/10.1007/978-1-4612-0011-6_13
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