Fuchsian equations in Sobolev spaces and blow-up
Résumé
We construct solutions of $\square u = e^u$ which blow-up precisely on a given space-like hypersurface of class $H^s$. 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.