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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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