Global Fuchsian Goursat problem in the class of holomorphic-gevrey functions

The first and the third authors have proved the global existence of holomorphic solutions to partial differential equations with several Fuchsian variables in the sense of N.S. Madi, under some assumptions on some coefficients and on the Fuchsian characteristic polynomial. This article shows a similar global existence of solutions which are holomorphic with respect to Fuchsian variables and of projective Gevrey class with respect to non-Fuchsian variables. The proof is based on the concept of formal norms of Leray-Waelbroeck and the application of Gevrey's operator of C. Wagschal.