Global Stability for Nonlinear Wave Equations Satisfying a Generalized Null Condition

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 Ck\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C<^>k$$\end{document} 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 rp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r<^>p$$\end{document} 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.