# zbMATH — the first resource for mathematics

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].

##### MSC:
 35L45 Initial value problems for first-order hyperbolic systems