## Cauchy problem for hyperbolic systems in Gevrey class. A note on Gevrey indices.(English)Zbl 1047.35086

The author considers the hyperbolic system $\begin{gathered} [I_4 D_t+ A(t) D_x+ B(t)]u(t,x)= 0,\\ u(0,x)= u_0(x)\end{gathered}$ in $$\Omega= [0,T]\times \mathbb{R}^1_x$$ where $$I_4$$ denotes the unit matrix of order 4 and $A(t)= \begin{pmatrix} \lambda(t) & 1 & 0 & 0\\ 0 &\lambda(t) & a(t) & 0\\ 0 & 0 & \mu(t) & 1\\ 0 & 0 & 0 & \mu(t)\end{pmatrix},$ $$\lambda(t)$$, $$\mu(t)$$, $$a(t)$$ are real smooth functions, with some assumptions.
The author determines completely the Gevrey indices for the well-posedness of the Cauchy problem; this proves that the maximal multiplicity of the zeros of the minimal polynomial of the principal part does not give, in general, the appropriate index for the Gevrey well posedness.

### MSC:

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

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