BLOW-UP SURFACES FOR NONLINEAR-WAVE EQUATIONS .1.

We introduce a systematic procedure for reducing nonlinear wave equati ons to characteristic problems of Fuchsian type. This reduction is com bined with an existence theorem to produce solutions blowing up on a p rescribed hypersurface. This first part develops the procedure on the example square u = exp(u); we find necessary and sufficient conditions for the existence of a solution of the form ln(2/phi2) + upsilon, whe re {phi = 0} is the blow-up surface, and upsilon is analytic. This giv es a natural way of continuing solutions after blow-up.