= 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 obeys
$$
\|F(w)\|_{L^2((0,T)\times U)}\leq C\sqrt T\,\|w\|_{X_T}^2,
$$
together 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>.
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