arXiv2010
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}ϕ-\partial_{x_i}\big(g^{ij}(t,x)\partial_{x_j}ϕ\big)+ϕ^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.