The [parabolic theory](/pages/1074) models diffusion — processes that smooth out irregularities and dissipate energy over time. Many physical systems, however, are governed by wave propagation: vibrating membranes, acoustic pressure, electromagnetic fields. These are described by **hyperbolic equations**, where the second-order time derivative $u_{tt}$ replaces the first-order $u_t$ of the parabolic case.

The archetype is the wave equation $u_{tt} - \Delta u = 0$: the displacement $u(x,t)$ of a vibrating medium evolves by oscillation rather than relaxation. More generally, a second-order hyperbolic equation takes the form

\begin{align*} u_{tt} + Lu = f \quad \text{in } U_T, \end{align*}

where $L$ is the same elliptic operator from the [elliptic](/pages/1033) and [parabolic](/pages/1074) theories. The sign convention $u_{tt} + Lu = f$ with $L = -\Delta$ gives the standard wave equation $u_{tt} - \Delta u = f$.

The passage from $u_t$ (parabolic) to $u_{tt}$ (hyperbolic) changes the character of the equation profoundly. Parabolic equations are infinite-dimensional analogues of $\dot{x} = Ax + b$; hyperbolic equations are analogues of $\ddot{x} = Ax + b$ — a Hamiltonian system. This shift has three fundamental consequences:

1. **Two initial conditions are needed**: position $u(\cdot, 0) = g$ and velocity $u_t(\cdot, 0) = h$, just as Newton's second law requires both initial position and velocity. 2. **Energy is conserved**, not dissipated. The total energy (kinetic + potential) remains constant in time. 3. **There is no smoothing effect**. Singularities in the initial data propagate at finite speed, rather than being instantly suppressed.

## Setup and Notation

As in the [parabolic theory](/pages/1074), we fix a bounded [open set](/page/Open%20Set) $U \subseteq \mathbb{R}^n$ with $C^1$ boundary, a time horizon $T > 0$, and the space-time cylinder $U_T := U \times (0, T]$. The elliptic operator $L$ in divergence form with time-dependent coefficients is exactly as before:

[definition: Hyperbolic Equation] Let $L$ be the second-order elliptic operator: \begin{align*} Lu(x, t) := -\sum_{i,j=1}^n \partial_{x_j}\bigl(a_{ij}(x,t)\,\partial_{x_i} u\bigr) + \sum_{i=1}^n b_i(x,t)\,\partial_{x_i} u + c(x,t)\,u, \end{align*} with $a_{ij}, b_i, c \in L^\infty(U_T)$ and uniform ellipticity: $\sum a_{ij}\xi_i\xi_j \ge \theta|\xi|^2$ for some $\theta > 0$. The **hyperbolic initial-[boundary](/page/Boundary) value problem** is: \begin{align*} \begin{cases} u_{tt} + Lu = f & \text{in } U_T, \\ u = 0 & \text{on } \partial U \times [0, T], \\ u(\cdot, 0) = g, \quad u_t(\cdot, 0) = h & \text{in } U. \end{cases} \end{align*} [/definition]

The initial data now consists of a **pair** $(g, h)$: the initial displacement $g$ and the initial velocity $h$. This is the essential structural difference from the parabolic problem, which requires only a single initial datum $g$. Physically, specifying $g$ alone determines the equilibrium (elliptic) or diffusion (parabolic) problem uniquely, but a wave requires both position and momentum to determine its future evolution.

## [Function](/page/Function) Spaces

The presence of $u_{tt}$ in the equation demands that the solution possess more temporal regularity than in the parabolic case. In the [parabolic theory](/pages/1074), the solution space $\mathcal{W}$ required only $u_t \in L^2(0,T; H^{-1}(U))$ — the time derivative could be a [distribution](/page/Distribution) in space. For hyperbolic equations, we need $u_t$ to be an honest $L^2$ function (reflecting the kinetic energy $\frac{1}{2}\|u_t\|_{L^2}^2$), and it is $u_{tt}$ that lives in $H^{-1}$.

[definition: The Hyperbolic Solution Space] A **weak solution** of the hyperbolic equation belongs to: \begin{align*} u \in L^2(0, T; H^1_0(U)), \qquad u_t \in L^2(0, T; L^2(U)), \qquad u_{tt} \in L^2(0, T; H^{-1}(U)). \end{align*} Equivalently, $u \in L^2(0,T; H^1_0(U))$ with $u' \in L^2(0,T; L^2(U))$ and $u'' \in L^2(0,T; H^{-1}(U))$, where primes denote distributional time [derivatives](/page/Derivative). [/definition]

[remark: Continuity in Time] As in the parabolic case, the regularity of $u$ and $u_t$ in time guarantees temporal [continuity](/page/Continuity): \begin{align*} u \in C([0, T]; H^1_0(U)), \qquad u_t \in C([0, T]; L^2(U)). \end{align*} The first ensures the initial displacement $u(\cdot, 0) = g$ is meaningful in $H^1_0(U)$; the second ensures the initial velocity $u_t(\cdot, 0) = h$ is meaningful in $L^2(U)$. Compare with the [parabolic case](/pages/1074), where $\mathcal{W} \hookrightarrow C([0,T]; L^2(U))$ — the hyperbolic solution has one more degree of temporal regularity, giving continuity into $H^1_0$ rather than $L^2$. [/remark]

## Weak Solutions

[definition: Weak Solution of the Hyperbolic Equation] Let $f \in L^2(0, T; L^2(U))$, $g \in H^1_0(U)$, and $h \in L^2(U)$. A function $u$ with the regularity above is a **weak solution** if: 1. **Weak PDE.** For every $v \in H^1_0(U)$ and a.e. $t \in (0, T)$: \begin{align*} u_{tt}(t) \circ v + B[u(t), v; t] = (f(t), v)_{L^2(U)}, \end{align*} where $\circ$ denotes the duality pairing between $H^{-1}(U)$ and $H^1_0(U)$, $B[\cdot, \cdot\,; t]$ is the bilinear form from the [parabolic theory](/pages/1074), and $(f, v)_{L^2}$ is the $L^2$ inner product (since $f \in L^2$, the right-hand side is an $L^2$ pairing rather than a duality pairing — the source term has more regularity than in the parabolic case). 2. **Initial conditions.** $u(\cdot, 0) = g$ in $H^1_0(U)$ and $u_t(\cdot, 0) = h$ in $L^2(U)$. [/definition]

[remark: Comparison With the Parabolic Weak Formulation] The parabolic weak formulation reads $u_t \circ v + B[u, v; t] = f \circ v$. The hyperbolic version replaces $u_t \circ v$ with $u_{tt} \circ v$ — the "forcing from the time direction" is now the acceleration rather than the velocity. The parabolic problem is first-order in time (one initial condition); the hyperbolic problem is second-order (two initial conditions). Note also that the source term $f$ is required to be in $L^2(0,T; L^2(U))$ for the hyperbolic theory, which is more regular than the $L^2(0,T; H^{-1}(U))$ allowed in the parabolic case. This reflects the fact that hyperbolic equations do not smooth: the data must be better to begin with. [/remark]

## Energy Estimates

The energy functional for hyperbolic equations has two components: kinetic energy $\frac{1}{2}\|u_t\|_{L^2}^2$ and potential energy encoded in the bilinear form. The key feature is that total energy is **conserved** (or at least bounded), in contrast to the parabolic case where it is **dissipated**.

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