Jannelli, Enrico Regularly hyperbolic systems and Gevrey classes. (English) Zbl 0583.35074 Ann. Mat. Pura Appl., IV. Ser. 140, 133-145 (1985). This paper deals with the first order Cauchy problem \[ (1)\quad \partial U/\partial t=\sum A_ h(t,x) \partial U/\partial x_ h+B(t,x),\quad U(0,x)=g(x), \] \(0\leq t\leq T\), \(x\in {\mathbb{R}}^ n\), where \(A_ h\) (1\(\leq h\leq n)\) and \(B\) are \(N\times N\) real matrices, while U and g are real \(N\)-vectors. System (1) is assumed to be regularly hyperbolic. Suppose that the coefficients \(A_ h(t,x)\) are Hölder continuous of order \(\alpha\) in t \((0<\alpha <1)\) and belong to the Gevrey class of order s in x and that \(B(t,x)\) is locally bounded and belongs to the Gevrey class of order s in x. Then the author proves that the Cauchy problem is well posed in the Gevrey class of order s provided that \(1\leq s<1/(1-\alpha)\). The method of energy inequalities is used. Reviewer: P.Jeanquartier Cited in 1 ReviewCited in 11 Documents MSC: 35L45 Initial value problems for first-order hyperbolic systems 35F10 Initial value problems for linear first-order PDEs 35R25 Ill-posed problems for PDEs 35L40 First-order hyperbolic systems 35L85 Unilateral problems for linear hyperbolic equations and variational inequalities with linear hyperbolic operators Keywords:Cauchy problem; regularly hyperbolic; Gevrey class; energy inequalities PDFBibTeX XMLCite \textit{E. Jannelli}, Ann. Mat. Pura Appl. (4) 140, 133--145 (1985; Zbl 0583.35074) Full Text: DOI References: [1] Colombini, F.; De Giorgi, E.; Spagnolo, S., Sur les équations hyperboliques avec des coefficients ne dépendent que du temps, Ann. Scuola Norm. Sup. Pisa, 6, 511-559 (1979) · Zbl 0417.35049 [2] Jannelli, E., Hyperbolic systems with coefficients analytic in space variables, Journal of Math. of Kyoto Un., 21, 4, 715-739 (1981) · Zbl 0478.35063 [3] Mizohata, S., Analyticity of solutions of hyperbolic systems with analytic coefficients, Comm. Pure Appl. Math., 14, 547-559 (1961) · Zbl 0105.07203 [4] Mizohata, S., The theory of partial differential equations (1973), Cambridge: University Press, Cambridge · Zbl 0263.35001 [5] Nishitant, T., Sur les équations hyperboliques à coefficients qui sont hölderiens en t et de la classe de Gevrey en x, Bull. de Sc. Math., 107, 2, 113-138 (1983) · Zbl 0536.35042 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.