 Solution

PutThe assumed energy estimate for the linear wave equation and the first nonlinear estimate give, for $w\in X_{b,\tau }$,Choose $b\ge 2C_{0}D$ and then choose $\tau >0$ so small that $C_0\beta {\tau }^{1/2}{b}^2\le b/2$. Then $A$ maps the closed ball $X_{b,\tau }$ into itself.

For $w,\tilde{w}\in X_{b,\tau }$, the difference $Aw-A\tilde{w}$ solves the linear equation with zero Cauchy data and forcing $F(w)-F(\tilde{w})$. The second nonlinear estimate therefore givesShrinking $\tau$ once more makes the coefficient strictly smaller than one. Since $X_{\tau }$ is a Banach space and its closed ball is complete, $A$ is a contraction on $X_{b,\tau }$.