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On the Cauchy problem for finitely degenerate hyperbolic equations of second order. (English) Zbl 1073.35145

The paper is devoted to following Cauchy problem for weakly hyperbolic second order operators with coefficients depending only on \(t\): \[ \partial^2_t u-\sum_{j,k=1}^na_{jk}(t)\partial_{x_j}\partial_{x_k}u + \sum_{j=1}^n b_j(t)\partial_{x_j} u=0,\qquad u(0,x)=u_0(x),\quad \partial_t u(0,x)=u_1(x), \tag{1} \] where \(a(t,\xi)=\sum_{j,k=1}^na_{jk}(t)\xi_j\xi_k/\xi^2\geq 0\) for all \(t\in {\mathbb R}\), \(\xi\in {\mathbb R}^n\). Under a precise hypothesis in which the finite order degeneracy of the symbol \(a(t,\xi)\) is combined with a Levi type condition on the terms of order one, a result of well-posedness in Gevrey spaces for (1) is obtained. This result is related to similar ones obtained by V. Ya. Ivrii [Sib. Math. J. 17, 921–931 (1977; Zbl 0404.35068)] and by H. Ishida and H. Odai [Funkc. Ekvacioj, Ser. Int 43, 71–85 (2000; Zbl 1142.35522)]. The proof is inspired to the technique of approximate energies developped by F. Colombini, E. De Giorgi and S. Spagnolo [Ann. Sc. Norm. Super. Pisa., Cl. Sci., IV. Ser. 6, 511–559 (1979; Zbl 0417.35049)].

MSC:

35L15 Initial value problems for second-order hyperbolic equations
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[1] [CDS]Colombini, F., De Giorgi, E. andSpagnolo, S., Sur les équations hyperboliques avec des coefficients qui ne dépendent que du temps,Ann. Scuola Norm. Sup. Pisa Cl. Sci. 6 (1979), 511–559. · Zbl 0417.35049
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