Propagation of wave front sets of solutions of the Cauchy problem for hyperbolic equations in Gevrey classes (Q1094587)

The main purpose of this work is to study the propagation of the wave front set in Gevrey classes \(WF_{G(\kappa)}u\) of the solutions of the Cauchy problem for hyperbolic equations with characteristics of variable multiplicities. The authors reduce this problem to the analysis of the first order pseudo-differential system \({\mathcal L}u(t)=0\), \(u(0)=f\) when the principal symbol of \({\mathcal L}\) is diagonal. They prove that the wave front \(WF_{G(\kappa)}u(t)\) is included in the set \(\Gamma (t,WF_{G(\kappa)}(f))\). Roughly speaking, the set \(\Gamma(t,V)\) is formed by the conic hull of the set of end points at \(\sigma =t\) of all \(\epsilon\)-admissible trajectories \(\gamma(\sigma)\) issued from \((y,\eta)\in V\). The \(\epsilon\)-admissible trajectories represent an \(\epsilon\)-approximation of the so called broken null-bicharacteristics. The proofs are based on a deep and careful examination of Fourier integral operators in Gevrey classes and on their multiproducts.