LAUSR

In the present work, we show that the wave equation in three-dimensional torus of power-type nonlinearity is ill posed around any (relatively) regular dynamics in the super-critical regime. In the cubic case, we can even choose the regular dynamics to be the smooth one. This result can serve as one motivation to consider initial datum as an “ensemble” to measure the size of the set of these “bad” datum in the sense of probability. This work also indicates that the probabilistic solutions constructed, for instance, in Burq–Tzvetkov [ 4] can only be well approximated by smooth solutions issued from certain regularized datum.