We introduce a systematic procedure for reducing nonlinear wave equations to characteristic problems of Fuchsian type. This reduction is combined with an existence theorem to produce solutions blowing up on a prescribed hypersurface. This first part develops the procedure on the example 2u = exp(u); we find necessary and sufficient conditions for the existence of a solution of the form ln(2/φ 2) + v, where {φ = 0} is the blow-up surface, and v is analytic. This gives a natural way of continuing solutions after blow-up.