Global Stability for Nonlinear Wave Equations Satisfying a Generalized Null Condition

Abstract

We prove global stability for nonlinear wave equations satisfying a generalized null condition. The generalized null condition is made to allow for null forms whose coefficients have bounded \(C^k\) norms. We prove both the pointwise decay and improved decay of good derivatives using bilinear energy estimates and duality arguments. Combining this strategy with the \(r^p\) estimates of Dafermos–Rodnianski then allows us to prove the global stability. The proof requires analyzing the geometry of intersecting null hypersurfaces adapted to solutions of wave equations.