We consider a class of Gowdy spacetimes that reduces the Einstein’s field equation into a system of two semilinear wave equations, by assuming a universe without matter in which the gravitational wave fronts repeat in space and are mutually parallel. To prove the existence of local solutions of the system, we adopt a functional setting to seek periodic-in-space solutions. We show that the spectrum of the linearization operator around the trivial solution are nonresonant eigenvalues. Then we discuss a small-divisor problem related to the application of the Lyapunov- Schmidt reduction. 

This ongoing research is a joint work with Hannes Stuke.