 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 $u\in X_{\tau }$ satisfieswith 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.