Global solutions of nonlinear wave equations in time dependent inhomogeneous media (Q394007)

In the paper there is treated the Cauchy problem to the equation \[ \frac{1}{\sqrt{-G}}\sum_{i,j=0}^{3}\partial _i(g^{i,j}\sqrt{-G}\partial _j\phi )=F(\phi ,\partial \phi ) \] on a Lorentzian manifold \((\mathbb{R}^{3+1},g)\) with small, smooth and compactly supported initial data, where \(G=\det(g)\) and a time dependent metric \(g\) coincides with the Minkowski metric outside of the cylinder \(K=\{(t,x)\: | x| \leq R\}\). If the nonlinearity \(F\) satisfies null condition outside of a large cylinder \(K\) and the metric \(g\) yields some assumptions (uniform hyperbolicity, ...), the author proves the existence of a unique global solution to the above problem and its energy and pointwise decay for \(t\rightarrow +\infty \).