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>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 105 / 3 / d / Solution

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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