Lemma: Smooth Cauchy problem on the slab

Let $\mu \ge 0$, $s\in C^{\infty }(M_T)$, $u_0,u_1\in C^{\infty }({\mathbb{R}}^3)$. There is a unique $u\in C^{\infty }(M_T)$ with

For $y\in M_T$, $u(y)$ depends only on the restrictions of $s,u_0,u_1$ to $J^{-}(y)\cap M_T$. If $s,u_0,u_1$ vanish outside a compact set $K_0$ and its causal future, then $\mathrm{supp}u\subset J^{+}(K_0)\cap M_T$. For every compact $K\subset M_T$ and every $N$ there are $N^{\prime }$ and $C$ with

Proof. For compactly supported smooth data and source, existence, uniqueness, the support bound, and continuous dependence are standard BGP (Chapter 3). Finite propagation for the constant-coefficient operator $\square +{\mu }^2$ is the energy argument of Evans (2.4.3), with ${\mu }^2{u}^2$ included in the energy density: a smooth solution of the homogeneous equation whose data vanish on $J^{-}(y)\cap \Sigma$ and whose source vanishes on $J^{-}(y)\cap M_T$ vanishes at $y$.

For general data, fix a compact $K\subset M_T$. The set ${\mathcal{D}}_K=J^{-}(K)\cap M_T$ is compact. Choose $\varrho \in {C_c}^{\infty }(M)$ equal to one on a neighbourhood of ${\mathcal{D}}_K$, and let $u_K$ be the compact-data solution for $(\varrho s,\varrho u_0,\varrho u_1)$. If ${\varrho }^{\prime }$ is another such cutoff, the difference $u_K-u_K^{\prime }$ has source and data vanishing on ${\mathcal{D}}_K$, hence vanishes on $K$ by finite propagation. Define $u=u_K$ on $K$. Compatibility over different $K$ follows in the same way, so $u$ is a well-defined smooth solution. Uniqueness and the dependence statement are finite propagation again. The estimate is the compact-case estimate applied to $(\varrho s,\varrho u_0,\varrho u_1)$.