Cauchy problem for semilinear wave equation with time-dependent metrics

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We establish the existence of weak solutions u of the semilinear wave equation ∂ₜ² u-divₓ(a(t,x)∇ₓu)=fₖ(u) where a(t,x) is equal to 1 outside a compact set with respect to x and a non-linear term fₖ which satisfies fₖ(u)≤ C uᵏ. For some non-trapping time-periodic perturbations a(t,x), we obtain the long time existence of solution for small initial data.