  We study the local existence of strong solutions for the cubic nonlinear wave
equation with data in $H^s(M)$, $s<1/2$, where $M$ is a three dimensional
compact riemannian manifold. This problem is supercritical and can be shown to
be strongly ill-posed (in the Hadamard sense). However, after a suitable
randomization, we are able to construct local strong solution for a large set
of initial data in $H^s(M)$, where $s\geq 1/4$ in the case of a boundary less
manifold and $s\geq 8/21$ in the case of a manifold with boundary.

Analysis of PDEs