Global stability of solutions to nonlinear wave equations

We consider the problem of global stability of solutions to a class of semilinear wave equations with null condition in Minkowski space. We give sufficient conditions on the given solution, which guarantees stability. Our stability result can be reduced to a small data global existence result for a class of semilinear wave equations with linear terms Bμν∂μΦ(t,x)∂νϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B^{\mu \nu }\partial _\mu \Phi (t, x)\partial _\nu \phi $$\end{document}, Lμ(t,x)∂μϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\mu (t,x)\partial _\mu \phi $$\end{document} and quadratic terms hμν(t,x)∂μϕ∂νϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h^{\mu \nu }(t, x)\partial _\mu \phi \partial _\nu \phi $$\end{document} where the functions Φ(t,x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi (t, x)$$\end{document}, Lμ(t,x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\mu (t, x)$$\end{document}, hμν(t,x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h^{\mu \nu }(t, x)$$\end{document} decay rather weakly and the constants Bμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B^{\mu \nu }$$\end{document} satisfy the null condition. We show the small data global existence result by using the new approach developed by Dafermos–Rodnianski. In particular, we prove the global stability result under weaker assumptions than those imposed by Alinhac (Indiana Univ Math J 58(6):2543–2574, 2009).