In this second part, we prove that the equation 2u = e u has solutions blowing up near a point of any analytic, space-like hypersurface in R n , without any additional condition; if (φ(x, t) = 0) is the equation of the surface, u − ln(2/φ 2) is not necessarily analytic, and generally contains logarithmic terms. We then construct singular solutions of general semilinear equations which blow-up on a non-characteristic surface, provided that the first term of an expansion of such solutions can be found. We finally list a few other simple nonlinear evolution equations to which our methods apply; in particular, formal solutions of soliton equations given by a number of authors can be shown to be convergent by this procedure.