Global existence of critical nonlinear wave equation with time dependent variable coefficients

In this paper, we establish global existence of smooth solutions for the Cauchy problem of the critical nonlinear wave equation with time dependent variable coefficients in three space dimensions {equation}\partial_{tt}\phi-\partial_{x_i}\big(g^{ij}(t,x)\partial_{x_j}\phi\big)+\phi^5=0, mathbb{R}_t \times \mathbb{R}_x^3,{equation} where $\big(g_{ij}(t,x)\big)$ is a regular function valued in the spacetime of $3\times3$ positive definite matrix and $\big(g^{ij}(t,x)\big)$ its inverse matrix. Here and in the sequence, a repeated sum on an index in lower and upper position is never indicated. In the constant coefficients case, the result of global existence is due to Grillakis \cite{Grillakis1}; and in the time-independent variable coefficients case, the result of global existence and regularity is due to Ibrahim and Majdoub \cite{Ibrahim}. The key point of our proofs is to show that the energy cannot concentrate at any point. For that purpose, following Christodoulou and Klainerman \cite{Chris}, we use a null frame associated to an optical function to construct a geometric multiplier similar to the well-known Morawetz multiplier. Then we use comparison theorem originated from Riemannian Geometry to estimate the error terms. Finally, using Strichartz inequality due to \cite{Smith} as Ibrahim and Majdoub \cite{Ibrahim}, we obtain global existence.