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.
Categories: math.AP