 Local weak solution by contraction for a semilinear wave equation

Suppose the inhomogeneous linear wave equation has an energy estimate in a Banach solution space $X_T$, and its nonlinearity obeystogether with the corresponding locally Lipschitz estimate. For a sufficiently large closed ball and sufficiently small $T$, the map that solves the linear equation with source $F(w)$ is a contraction. The contraction mapping theorem then gives a local weak solution.