We show how the Lasserre hierarchy can solve a class of non-linear partial differential equations (PDEs), with rigorous convergence guarantees. We use a weak formulation of the nonlinear PDE, resulting in a linear equation in the space of measures, to be solved numerically and approximately with the hierarchy. The entire approach is based purely on convex optimization and it does not rely on spatio-temporal gridding, even though the PDE addressed can be fully nonlinear.