Asymptotic expansions of solutions of Fuchsian hyperbolic equations in spaces of functions of Gevrey classes (Q1075525)

This note deals with Fuchsian hyperbolic equations of the type \[ (E)\quad (t\partial_ t)^{m_ u}+\sum t^{q(j,\alpha)}a_{j,\alpha}(t,x)(t\partial_ t)^ j\partial_ x^{\alpha}u=0\quad, \] where \(m\geq 1\), \(j+| \alpha | \leq m\), \(j<m\), \(0\leq t\leq T\), \(x\in {\mathbb{R}}^ n\); the q(j,\(\alpha)\)'s are suitable nonnegative integers. The coefficients \(a_{j,\alpha}(t,x)\) are \(C^{\infty}\) functions of t with values in the Gevrey class \(G^ s({\mathbb{R}}^ n)\) of index s; s is an appropriate real number \(>1\). The characteristic exponents \(\rho_ 1(x),...,\rho_ m(x)\) are the roots of \[ \rho^ m+\sum_{0\leq j<m}\rho^ j[t^{q(j,0)}a_{j,0}(t,x)]_{t=0}; \] one assumes that \(\rho_ i(x)- \rho_ j(x)\) is not an integer for \(i\neq j\) and \(x\in {\mathbb{R}}^ n\). The following results are given without proof: 1) any solution u(t,x) of (E) which is a \(C^{\infty}\) function of t with values in \(G^ s({\mathbb{R}}^ n)\) can be expanded asymptotically into the form \[ (A)\quad u(t,x)\sim \sum^{m}_{i=1}(f_ i(x)t^{\rho_ i(x)}+\sum^{\infty}_{k=1}\sum^{mk}_{\quad h=0}f_{i,k,h}(x)t^{\rho_ i(x)+k} (\log t)^{mk-h})\quad as\quad t\to +0 \] for some unique \(f_ i\), \(f_{i,k,h}\in G^ s({\mathbb{R}}^ n)\); 2) conversely, for any \(f_ 1,...,f_ m\in G^ s({\mathbb{R}}^ n)\) there exist a unique solution u of (E) and unique coefficients \(f_{i,k,h}\) such that the asymptotic relation (A) holds.