$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$
Consider multidimensional wave equation \begin{equation} u_{tt}-c^2 \Delta u=0. \label{eq-2.7.1} \end{equation} Recall that $\Delta =\nabla\cdot \nabla=\partial_x^2+\partial_y^2+\partial_z^2$ (the number of terms depends on dimension). Multiplying by $u_t$ we get in the left-hand expression \begin{align*} u_t u_{tt}-c^2 u_t\nabla^2 u= &\partial_t \bigl(\frac{1}{2}u_t^2 \bigr)+ \nabla \cdot (-c^2 u_t \nabla u )+ c^2 \nabla u_t \cdot \nabla u \\ =&\partial_t \bigl(\frac{1}{2}u_t^2 + \frac{1}{2}c^2 |\nabla u|^2\bigr)+ \nabla \cdot \bigl(-c^2 u_t \nabla u \bigr). \end{align*} So we arrive to \begin{equation} \partial_t \bigl(\frac{1}{2}u_t^2 + \frac{1}{2}c^2 |\nabla u|^2\bigr)+ \nabla \cdot \bigl(-c^2 u_t \nabla u \bigr). \label{eq-2.7.2} \end{equation} This is energy conservation law in the differential form. Here \begin{equation} e= \bigl(\frac{1}{2}u_t^2 + \frac{1}{2}c^2 |\nabla u|^2\bigr) \label{eq-2.7.3} \end{equation} is a density of energy and \begin{equation} \mathbf{S}=-c^2 u_t \nabla u \label{eq-2.7.4} \end{equation} is a vector of energy flow.
Then if we fix a volume (or an area in $2D$ case, or just an interval in $1D$ case) $V$ and introduce a full energy in $V$ at moment $t$ \begin{equation} E_V(t)= \iiint _V \bigl(\frac{1}{2}u_t^2 + \frac{1}{2}c^2 |\nabla u|^2\bigr)\, dV \label{eq-2.7.5} \end{equation} then \begin{equation} E_V(t_2) - E_V(t_1) + \int_{t_1}^{t_2}dt \iint_\Sigma \mathbf{S}\cdot \mathbf{n}\,d\sigma =0 \label{eq-2.7.6} \end{equation} where $\Sigma$ is the surface bounding $V$, $d\sigma$ is an element of the surface area, and $\mathbf{n}$ is an unit exterior normal to $\Sigma$.
Similarly, Maxwell equations without charges and currents are \begin{align} &\varepsilon \mathbf{E}_t = c\nabla \times \mathbf{H},\label{eq-2.7.7}\\ &\mu\mathbf{H}_t = -c\nabla \times \mathbf{E},\label{eq-2.7.8}\\ &\nabla \cdot \varepsilon\mathbf{E}=\nabla \cdot \mu\mathbf{H}=0. \label{eq-2.7.9} \end{align} Here $\mathbf{E}, \mathbf{H}$ are intensities of electric and magnetic field respectively, $c$ is the speed of light in the vacuum, $\varepsilon$ and $\mu$ are dielectric and magnetic characteristics of the material ($\varepsilon\ge 1$, $\mu \ge 1$ and $\varepsilon=\mu=1$ in the vacuum).
Multiplying (taking an inner product) (\ref{eq-2.7.7}) by $\mathbf{E}$ and (\ref{eq-2.7.8}) by $\mathbf{H}$ and adding we arrive to \begin{equation*} \partial_t \bigl(\frac{1}{2}\varepsilon|\mathbf{E}|^2 + \frac{1}{2}\mu |\mathbf{H}|^2\bigr)= c\bigl(\mathbf{E}\cdot (\nabla \times \mathbf{H}) - \mathbf{H}\cdot (\nabla \times \mathbf{E})\bigr)= -c\nabla \cdot \bigl(\mathbf{E}\times \mathbf{H}\bigr) \end{equation*} where the last equality follows from vector calculus.
Then \begin{equation} \partial_t \bigl(\frac{1}{2}\varepsilon |\mathbf{E}|^2 + \frac{1}{2}\mu |\mathbf{H}|^2\bigr)+ \nabla \cdot \bigl(c\mathbf{E}\times \mathbf{H}\bigr)=0. \label{eq-2.7.10} \end{equation} In the theory of electromagnetism \begin{equation} e=\frac{1}{2}\bigl(\varepsilon |\mathbf{E}|^2 + \mu |\mathbf{H}|^2\bigr) \label{eq-2.7.11} \end{equation} is again density of energy and \begin{equation} \mathbf{S}=c \mathbf{E}\times \mathbf{H} \label{eq-2.7.12} \end{equation} is a vector of energy flow (aka Poynting vector).
Remark 1. $\frac{c}{\sqrt{\mu \varepsilon}}$ is the speed of light in the given material.
Remark 2. In inhomogeneous material $\varepsilon$ and $\mu$ depend on $(x,y,z)$; in anisotropic material (f.e. crystal) $\varepsilon$ and $\mu$ are symmetric matrices and then $e=\frac{1}{2}\bigl(\varepsilon \mathbf{E} \cdot \mathbf{E} + \mu \mathbf{H}\cdot \mathbf{H}\bigr)$.
Elasticity equations in homogeneous isotropic material are \begin{equation} \mathbf{u}_{tt}= 2\mu \Delta \mathbf{u} + \lambda \nabla (\nabla \cdot \mathbf{u}) \label{eq-2.7.13} \end{equation} where $\mathbf{u}$ is a displacement and $\lambda>0$, $\mu>0$ are Lamé parameters.
Problem 1. Multiplying (taking an inner product) (\ref{eq-2.7.13}) by $\mathbf{u}_t$ write conservation law in the differential form. What are $e$ and $\mathbf{S}$?
More examples of the conservation laws in the differential form could be found in Problems to Section 2.7 and Section 14.1
Energy integral can be used to prove the uniqueness of the solution (see Section 9.2) and, combined with the functional-analytical methods (which means methods of Analysis III a.k.a Real Analysis) to prove also the existence and the well-posedness.