Cauchy problems for Fuchsian hyperbolic equations in spaces of functions of Gevrey classes (Q1071952)

This note deals with the Cauchy problem \[ t^ k \partial^ m_ t u+\sum t^{p(j,\alpha)} a_{j,\alpha} \partial^ j_ t \partial_ x^{\alpha} u=f(t,x),\quad \partial^ i_ t u(0,x)=u_ i(x), \] i\(=0,1,...,m-k-1\), \(0\leq t\leq T\), \(x\in {\mathbb{R}}^ n\), where m, k are integers, \(0\leq k<m\), \(0\leq p(j,\alpha)\) suitable integers, \(j+| \alpha | \leq m\), \(j<m\), with suitable hyperbolicity conditions. The problem is well posed in spaces of \(C^{\infty}\) functions and in spaces of functions of Gevrey classes. Moreover the solution has finite propagation speed.