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

The condition ${\sum }_{i=1}^{n}1/p_i=1$ and positivity imply $p_i\ge 1$. We prove the Generalized Holder inequality by induction on $n$. The case $n=2$ is the usual Holder inequality. For the induction step, setThe exponents $q$ and $p_n$ are conjugate, so Hölder followed by the induction hypothesis givesThis proves the claim.

Since $U\subset {\mathbb{R}}^3$ is bounded and smooth, the Sobolev embedding theorem gives continuous embeddings $H^1(U)\hookrightarrow L^4(U)$ and $H^1(U)\hookrightarrow L^6(U)$. Because $|\sin s|\le 1$,Taking the $L^2$ norm in time gives

For $F(s)=s^2\sin s$, the mean value theorem and $|F^{\prime }(s)|\le 2|s|+s^2$ implyApply the Generalized Holder inequality in space, using $L^4$ for the quadratic products and $L^6\cdot L^3$ for the cubic products, and then use the two Sobolev embeddings. Pointwise in time this yieldsTaking the $L^2$ norm in time proves the required estimate with the factor $\gamma T^{1/2}$.

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 $X_{b,\tau }$.

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.