Global well-posedness for the nonlinear wave equation with a cubic nonlinearity in two space dimensions

In this paper, we prove the global well-posedness for the nonlinear wave equation with a cubic nonlinearity utt−Δu=−u3utt−Δu=−u3 in two space dimensions, with initial data (u0,u1)∈Ḣxs×Ḣxs−1, 1>s>511≃0.4545.