Positive matrix-potential wave system energy

Let $N\ge 1$, $I:=[0,T)$ with $0<T\le \mathrm{\infty }$, and let $J\subseteq \mathbb{R}$ be an open interval containing $I$. Let $U,S\in C^{\mathrm{\infty }}(J\times \mathbb{R},{\mathbb{R}}^N)$. Write ${\mathrm{Sym}}_N(\mathbb{R})$ for the real symmetric $N\times N$ matrices and use the Euclidean dot product and norm on ${\mathbb{R}}^N$. Let $V\in C^{\mathrm{\infty }}(\mathbb{R},{\mathrm{Sym}}_N(\mathbb{R}))$ be bounded, independent of $t$, and positive semidefinite at every $x$, meaning $z^{\mathsf{T}}V(x)z\ge 0$ for every $z\in {\mathbb{R}}^N$. Assume $U$ and all its mixed derivatives of total order at most two, and $S$, are Schwartz in $x$ locally uniformly for $t\in J$: for every compact $K\subset J$, every such derivative $D$ and every integer $m\ge 0$, ${\sup }_{t\in K,x\in \mathbb{R}}(1+\mathrm{\mid }x\mathrm{\mid }{)}^m\mathrm{\mid }DU(t,x)\mathrm{\mid }<\mathrm{\infty }$, and the analogous bounds hold for $S$. If $U_{tt}-U_{xx}+V(x)U=S,$ define $E(t):=\frac{1}{2}{\int }_{\mathbb{R}}(\mathrm{\mid }{U}_t{\mathrm{\mid }}^2+\mathrm{\mid }{U}_x{\mathrm{\mid }}^2+U^{\mathsf{T}}V(x)U)dx,$ where $\mathrm{\mid }\cdot \mathrm{\mid }$ is the Euclidean norm and $\mathrm{\parallel }v(t,\cdot ){\mathrm{\parallel }}_2^2:={\int }_{\mathbb{R}}\mathrm{\mid }v(t,x){\mathrm{\mid }}^{2}dx$ for any vector-valued function $v$. Then $E(t)\ge 0$, $E^{\mathrm{\prime }}(t)={\int }_{\mathbb{R}}{U}_t\cdot Sdx,\sqrt{E(t)}\le \sqrt{E(0)}+\frac{1}{\sqrt{2}}{\int }_0^t\mathrm{\parallel }S(\tau ,\cdot ){\mathrm{\parallel }}_{2}d\tau (t\in I)\mathrm{.}$ In particular $E$ is constant when $S=0$. If $t,x$ have length units and $U$ has unit $[U]$, then $V$ has unit ${\mathrm{m}}^{-2}$, $S$ has unit $[U]{\mathrm{m}}^{-2}$, $E$ has unit $[U{]}^2{\mathrm{m}}^{-1}$, and both sides of the square-root estimate have unit $[U]{\mathrm{m}}^{-1\mathrm{/}2}$. This abstract estimate supplies neither pointwise decay nor a reconstruction of coupled Einstein–Maxwell perturbations.

The estimate is finite-time energy control, not decay. To use it for an actual coupled perturbation, one must first derive that system's exact variables, angular sectors, symmetric structure, and reconstruction.