In the Blair, Smith and Sogge's paper Strichartz estimates for the wave equation on manifolds with boundary, the authors study integrability estimates for solution of the following problem:

\begin{equation} \partial_t^2 u(t,x)-\Delta_g u(t,x)=0, ~ u(0,x)=f(x), ~ \partial_tu(0,x)=g(x) \tag{1.1} \end{equation} where $\Delta_g$ denotes the Laplace–Beltrami operator on $(M,g)$ with $(M, g)$ a Riemannian manifold of dimension $n \geq 2$.

They establish the following result:

Theorem 1.1. Let $M$ be a compact Riemannian manifold with boundary. Suppose $2<p\leq \infty$, $2 \leq q < \infty$ and $(p,q,\gamma)$ is a triple satisfying \begin{equation} \frac{1}{q} + \frac{n}{q}= \frac{n}{2}-\gamma \hspace{ 2cm}\begin{cases} \frac{3}{p}+\frac{n-1}{q} \leq \frac{n-1}{2}, n \leq 4 \\ \frac{1}{p}+\frac{1}{q} \leq \frac{1}{2}, n \geq 4 \end{cases} \end{equation} Then we have the following estimates for solutions $u$ to (1.1) satisfying either Dirichlet or Neumann homogeneous boundary conditions \begin{equation} \|u\|_{L^{p}([-T,T];L^q(M))}\leq C(\|f\|_{H^{\gamma}(M)}+\|g\|_{H^{\gamma-1}(M)})\end{equation} with $C$ some constant depending on $M$ and $T$.

My question: Does anyone know of any place that has a didactic proof (with details) of this result, showing the dependency of the constant $C$ on $T$ and $M$? Containing the prerequisites and tools necessary to understand the proof.

The proof can be done in a bounded domain of $\mathbb{R}^{n}$.


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