Cauchy problem for semilinear wave equation with time-dependent metrics

Abstract

We establish the existence of a weak solution u of the semilinear wave equation ∂ t 2 u − d i v x ( a ( t , x ) ∇ x u ) = f k ( u ) where a ( t , x ) is equal to 1 outside a compact set with respect to x and a non-linear term f k which satisfies | f k ( u ) | ≤ C | u | k . For some non-trapping time-periodic perturbations a ( t , x ) , we obtain the long time existence of a solution from little initial data.