Fuchsian equations in Sobolev spaces and blow-up

International audienceWe construct solutions of $\square u = e^u$ which blow-upprecisely on a given space-like hypersurface of class $H^s$. Forthis purpose, we prove a general existence theorem for FuchsianPDE in Sobolev spaces. The precise relation between the regularityof the data and that of the solution is shown to involvelogarithmic symbols, in a model situation. A few further resultson power nonlinearities are also included