We will discuss a complementary result to the probabilistic well-posedness for the nonlinear wave equation. More precisely, we show that there is a dense G_delta 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 an approximate identity of the elements of S, does not converge in the space of super-critical Sobolev regularity.  This is a joint work with Chenmin Sun.