  We prove a complementary result to the probabilistic well-posedness for the
nonlinear wave equation. More precisely, we show that there is a dense set $S$
of the Sobolev space of super-critical regularity such that (in sharp contrast
with the probabilistic well-posedness results) the family of global smooth
solutions, generated by the convolution with some approximate identity of the
elements of $S$, does not converge in the space of super-critical Sobolev
regularity.
