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On the Cauchy problem for hyperbolic operators with non-regular coefficients. (English) Zbl 1036.35122
de Gosson, Maurice (ed.), Jean Leray ’99 conference proceedings. The Karlskrona conference, Sweden, August 1999 in honor of Jean Leray. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-1378-7/hbk). Math. Phys. Stud. 24, 37-52 (2003).
This paper deals with the well posedness of the Cauchy problem for a class of strictly hyperbolic operators having non-regular coefficients that depend only on the time variable \(t\). These are the main conditions imposed on the coefficients \(a_{ij}(t)\) of the principal part \(a(t,\xi)\) of the operator: \((*)\) there exist \(\overline t\in [0,T]\), \(q\), \(C>0\) and such that \(a_{ij}\in C^1([0, T]\setminus\{\overline t\})\) and \(|\partial_t a(t,\xi)|\leq C| t-\overline t|^{-q}\), for all \(t\in [0,T]\setminus\{\overline t\}\) and \(|\xi|= 1\).
Assuming that \(q= 1\) it is proved that the corresponding Cauchy problem is \(C^\infty\) well posed, while the assumption \(1< q< 2\) in \((*)\) implies the well posedness of the Cauchy problem in the Gevrey space \(\gamma^{(s)}\) for all \(s< {1\over q-1}\). The above formulated results are sharp.
For the entire collection see [Zbl 1017.00056].

35L45 Initial value problems for first-order hyperbolic systems