= Solution
By the <contraction mapping theorem>, the contraction $A:X_{b,\tau}\to X_{b,\tau}$ has a unique fixed point $u$. The identity $Au=u$ says exactly that $u$ is the weak solution of the linear initial boundary value problem whose forcing is $u^2\sin u$. Consequently $u\in X_\tau$ satisfies
$$
u_{tt}-\Delta u=u^2\sin u
$$
with the prescribed initial and homogeneous Dirichlet data. Thus a sufficiently small $\tau>0$ gives a local <weak solution> of the <semilinear wave equation>, as summarized by the <local weak solution by contraction for a semilinear wave equation>.
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