On the Gevrey strong hyperbolicity (Q2360869)

The author considers a weakly hyperbolic operator \[ P= D^m_0+ \sum_{\substack{ |\alpha|= m\\ \alpha_0< m}} a_\alpha(x)\,D^\alpha, \] \(x=(x_0,x')= (x_0,x_1,\dots, x_n)\), with Gevrey regular coefficients. Precise conditions on the multiple characteristics are given, granting the \(s\)-Gevrey well-posedness of the Cauchy problem \[ \begin{gathered} Pu+Qu= 0,\\ D^j_0 u(0,x')= u_j(x'),\;j=0,1,\dots,m-1,\end{gathered} \] for arbitrary lower order perturbations \(Q\) and any \(1<s<m/(m-2)\). Let us recall that, under the only assumption of weak hyperbolicity, we have \(s\)-Gevrey well-posedness for arbitrary lower order \(Q\) and \(1<s<m/(m-1)\), see \textit{M. D. Bronshtein} [Tr. Mosk. Mat. O.-va 41, 83--99 (1980; Zbl 0468.35062)]. Several examples are given and the doubly characteristic case is studied in detail.