On the Cauchy problem for finitely degenerate hyperbolic equations of second order (Q1426910)

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 \textit{V. Ya. Ivrii} [Sib. Math. J. 17, 921--931 (1977; Zbl 0404.35068)] and by \textit{H. Ishida} and \textit{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 \textit{F. Colombini}, \textit{E. De Giorgi} and \textit{S. Spagnolo} [Ann. Sc. Norm. Super. Pisa., Cl. Sci., IV. Ser. 6, 511--559 (1979; Zbl 0417.35049)].