Fuchsian Equations in Sobolev Spaces and Blow-Up

We construct solutions of □u=euwhich blow-up precisely on a given space-like hypersurface of classHs. For this purpose, we prove a general existence theorem for Fuchsian PDE in Sobolev spaces. The precise relation between the regularity of the data and that of the solution is shown to involve logarithmic symbols, in a model situation. A few further results on power nonlinearities are also included