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**. {width=80%} [theorem: Hyperbolic Energy Estimate] Let $u$ be a weak solution of $u_{tt} + Lu = f$ with $u(\cdot, 0) = g$ and $u_t(\cdot, 0) = h$, where the symmetric part of $L$ is uniformly elliptic. There exists a constant $C > 0$, depending only on $U$, $T$, $\theta$, and the $L^\infty$ norms of the coefficients, such that: \begin{align*} \max_{0 \le t \le T}\bigl(\|u_t(t)\|_{L^2(U)}^2 + \|u(t)\|_{H^1_0(U)}^2\bigr) \le C\left(\|h\|_{L^2(U)}^2 + \|g\|_{H^1_0(U)}^2 + \int_0^T \|f(t)\|_{L^2(U)}^2\,dt\right). \end{align*} [/theorem] [proof] **Step 1: Testing with $u_t$.** Set $v = u_t(t) \in L^2(U) \subset H^{-1}(U)$ in the weak formulation. Since $u_t \in L^2(0,T; L^2(U))$ and we need $v \in H^1_0(U)$, this requires a density argument (first prove the estimate for smooth approximations, then pass to the [limit](/page/Limit)). Proceeding formally: \begin{align*} u_{tt} \circ u_t + B[u, u_t; t] = (f, u_t)_{L^2}. \end{align*} **Step 2: Time derivatives of the energy components.** The first term gives: \begin{align*} u_{tt} \circ u_t = \frac{1}{2}\frac{d}{dt}\|u_t(t)\|_{L^2}^2. \end{align*} For the bilinear form, when $L$ is symmetric (i.e. $b_i = 0$ and $a_{ij} = a_{ji}$) with time-independent coefficients: \begin{align*} B[u, u_t; t] = \frac{1}{2}\frac{d}{dt}B[u, u; t]. \end{align*} In the general case with time-dependent coefficients or lower-order terms, there are correction terms controlled by $\|u\|_{H^1_0}^2$. **Step 3: Energy inequality.** Defining the energy: \begin{align*} e(t) := \frac{1}{2}\|u_t(t)\|_{L^2}^2 + \frac{1}{2}B[u(t), u(t); t], \end{align*} the computation gives (in the symmetric, time-independent coefficient case): \begin{align*} e'(t) = (f(t), u_t(t))_{L^2}. \end{align*} By Cauchy-Schwarz: $|e'(t)| \le \|f(t)\|_{L^2}\|u_t(t)\|_{L^2}$. In the general case, additional terms are bounded by Gronwall-type arguments. Integrating and applying Gronwall's inequality yields the stated estimate. [/proof] [remark: Energy Conservation for the Wave Equation] When $L = -\Delta$ (no lower-order terms), $f = 0$, and the coefficients are time-independent, the energy is exactly conserved: \begin{align*} E(t) := \frac{1}{2}\|u_t(t)\|_{L^2(U)}^2 + \frac{1}{2}\|\nabla u(t)\|_{L^2(U)}^2 = E(0) \quad \text{for all } t \in [0, T]. \end{align*} The first term $\frac{1}{2}\|u_t\|_{L^2}^2$ is the **kinetic energy** and the second $\frac{1}{2}\|\nabla u\|_{L^2}^2$ is the **potential energy**. These oscillate back and forth (as the wave alternates between moving fast with small displacement and being nearly stationary with large displacement), but their sum is constant. This is in stark contrast to the [parabolic energy identity](/pages/1074), where the energy $\frac{1}{2}\|u(t)\|_{L^2}^2$ is strictly decreasing and the gradient term $\int_0^t \|\nabla u\|_{L^2}^2\,ds$ appears as a cumulative dissipation. [/remark] As before, the energy estimate immediately implies uniqueness: if $u_1$ and $u_2$ are both weak solutions with the same data $(f, g, h)$, then $w = u_1 - u_2$ satisfies $w_{tt} + Lw = 0$ with $w(\cdot, 0) = 0$ and $w_t(\cdot, 0) = 0$. The estimate gives $\|w_t\|_{L^2}^2 + \|w\|_{H^1_0}^2 = 0$ for all $t$, so $w \equiv 0$. ## Existence of Weak Solutions: Galerkin's Method The Galerkin method for hyperbolic equations follows the same five-step strategy as in the [parabolic case](/pages/1074), with one crucial difference: the finite-dimensional ODE system is **second-order** rather than first-order. [motivation] ### The Strategy 1. **Choose a basis.** Let $\{w_k\}_{k=1}^\infty$ be an orthonormal basis of $L^2(U)$ consisting of eigenfunctions of the symmetric part of $L$, so that $\{w_k\}$ is also orthogonal in $H^1_0(U)$. 2. **Finite-dimensional approximation.** For each $m$, seek: \begin{align*} u_m(x, t) := \sum_{k=1}^m d_m^k(t)\,w_k(x). \end{align*} 3. **Second-order ODE system.** Testing the equation $(u_m)_{tt} \circ w_k + B[u_m, w_k; t] = (f, w_k)_{L^2}$ gives: \begin{align*} \ddot{d}_m^k(t) + \sum_{l=1}^m B[w_l, w_k; t]\,d_m^l(t) = (f(t), w_k)_{L^2}, \qquad k = 1, \ldots, m. \end{align*} This is a system of $m$ **second-order** linear ODEs (compare with the first-order system $\dot{d}_m^k + \cdots = \cdots$ in the parabolic case). By standard ODE theory, there exists a unique solution on $[0, T]$ with initial conditions $d_m^k(0) = (g, w_k)_{L^2}$ and $\dot{d}_m^k(0) = (h, w_k)_{L^2}$. 4. **Energy estimates.** Test the ODE system with $\dot{d}_m^k(t)$ (equivalently, set $v = (u_m)_t$ in the weak formulation). The same argument as in the energy estimate theorem gives uniform bounds on $\|(u_m)_t\|_{L^2}$ and $\|u_m\|_{H^1_0}$, independent of $m$. 5. **Passage to limit.** The uniform bounds give [weak convergence](/page/Weak%20Convergence) of a subsequence. Since the equation is linear, passing to the limit in the weak formulation is straightforward (no need for the Aubin-Lions strong compactness used in the parabolic case — the linearity of all terms suffices). [/motivation] [theorem: Existence of Weak Solutions] Let $L$ be uniformly elliptic with $L^\infty$ coefficients on $U_T$, let $f \in L^2(0, T; L^2(U))$, $g \in H^1_0(U)$, and $h \in L^2(U)$. Then there exists a unique weak solution $u$ of: \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*} with $u \in L^2(0,T; H^1_0(U))$, $u_t \in L^2(0,T; L^2(U))$, $u_{tt} \in L^2(0,T; H^{-1}(U))$. Moreover, $u$ satisfies the energy estimate. [/theorem] ## Regularity The regularity theory for hyperbolic equations differs fundamentally from the parabolic case. The key principle is: **regularity is preserved but not gained**. If the initial data $(g, h)$ has $k$ spatial derivatives, the solution has $k$ spatial derivatives for all time — but never more. {width=80%} [theorem: Hyperbolic Regularity] Let $u$ be the weak solution of $u_{tt} + Lu = f$, $u(\cdot,0) = g$, $u_t(\cdot,0) = h$ on $U_T$. Suppose the coefficients of $L$ and the boundary $\partial U$ are $C^\infty$. If $f, f_t, \ldots, f^{(m-1)} \in L^2(0,T; L^2(U))$, $g \in H^{m+1}(U)$, and $h \in H^m(U)$ (with appropriate compatibility conditions at $\partial U \times \{0\}$), then: \begin{align*} u \in C^k([0, T]; H^{m+1-k}(U)) \quad \text{for } k = 0, 1, \ldots, m+1. \end{align*} [/theorem] The pattern is: each time derivative "costs" one spatial derivative. The solution has $m+1$ spatial derivatives (from $g \in H^{m+1}$) at the expense of zero time derivatives; it has $m$ spatial derivatives and one time derivative (from $h \in H^m$); and so on down to $m+1$ time derivatives with zero spatial derivatives. The total count of derivatives (spatial + temporal) is bounded by $m+1$. [remark: No Smoothing — Contrast With Parabolic Equations] The hyperbolic regularity theorem should be compared directly with the [parabolic regularity theory](/pages/1074). For the [heat equation](/page/Heat%20Equation) with $g \in L^2(U)$, the solution becomes $C^\infty$ instantly for $t > 0$ — an infinite gain of regularity. For the wave equation with $g \in H^1(U)$ and $h \in L^2(U)$, the solution stays in $H^1$ for all time and never becomes smoother. The physical reason is clear: heat diffusion smooths by mixing nearby values, rapidly averaging out any sharp features. Waves, by contrast, transport disturbances without mixing — a sharp wavefront remains sharp as it propagates. Mathematically, the difference traces to the spectral behaviour. In the eigenfunction expansion for the heat equation, the coefficient of $w_k$ decays as $e^{-\lambda_k t}$, which decays faster for higher frequencies. In the wave equation, the coefficient oscillates as $\cos(\sqrt{\lambda_k}\,t)$, which has unit amplitude for all frequencies — no suppression of high-frequency modes occurs. [/remark] ## Finite Speed of Propagation Perhaps the most physically important distinction between parabolic and hyperbolic equations is the **speed of propagation of disturbances**. For the heat equation, a localised initial disturbance is instantly felt everywhere in the domain (infinite speed of propagation). For the wave equation, disturbances travel at a finite speed determined by the coefficients. {width=80%} [theorem: Finite Speed of Propagation] Consider the wave equation $u_{tt} - \Delta u = 0$ in $\mathbb{R}^n \times (0, \infty)$ with initial data $u(\cdot, 0) = g$ and $u_t(\cdot, 0) = h$ supported in the ball $B(x_0, r)$. Then: \begin{align*} u(x, t) = 0 \quad \text{for } |x - x_0| > r + t. \end{align*} More generally, the solution $u(\cdot, t)$ is supported in the ball $B(x_0, r + t)$. [/theorem] The proof uses the energy estimate applied to the **backward light cone**: define $e_R(t) := \frac{1}{2}\int_{B(x_0, R-t)} (u_t^2 + |\nabla u|^2)\,d\mathcal{L}^n$ for $R > r$. A calculation shows that $e_R(t)$ is non-increasing in $t$ (the energy flux through the lateral surface of the cone is non-positive). Since $e_R(0) = 0$ when $R > r$ (the initial data is supported in $B(x_0, r)$), it follows that $e_R(t) = 0$ for all $t$, giving $u \equiv 0$ outside the forward cone. For a general uniformly elliptic operator with bounded coefficients, the speed of propagation is finite but depends on the ellipticity constant $\theta$ and the $L^\infty$ bounds on the coefficients. The domain of influence of a point $(x_0, 0)$ is contained in a cone $\{(x, t) : |x - x_0| \le ct\}$ for some finite speed $c > 0$. [remark: No Maximum Principle] A significant absence in the hyperbolic theory is a **maximum principle**. For [elliptic](/pages/1033) and [parabolic](/pages/1074) equations, the maximum principle provides pointwise bounds: solutions are controlled by their boundary (or initial-boundary) data. No such principle holds for [wave equations](/page/Wave%20Equation). The physical reason is immediate: a vibrating string can overshoot its initial and boundary values. If $u(x,0) = 0$ and $u_t(x,0) = h(x) > 0$, the string starts at rest and moves upward — the solution $u(x,t) > 0$ for small $t > 0$, exceeding the initial value $0$ everywhere. The energy-based estimates replace the maximum principle as the fundamental tool for pointwise-type control in the hyperbolic theory. [/remark] ## Examples [example: The Wave Equation on a Bounded Domain] Consider the wave equation with homogeneous Dirichlet conditions: \begin{align*} \begin{cases} u_{tt} - \Delta u = 0 & \text{in } U \times (0, \infty), \\ u = 0 & \text{on } \partial U \times [0, \infty), \\ u(\cdot, 0) = g, \quad u_t(\cdot, 0) = h & \text{in } U. \end{cases} \end{align*} Let $\{\lambda_k, w_k\}$ be the eigenvalues and $L^2$-orthonormal eigenfunctions of $-\Delta$ on $U$ with Dirichlet conditions (as in the [elliptic spectral theory](/pages/1033)). The Galerkin ODE for the $k$-th mode is $\ddot{d}^k + \lambda_k d^k = 0$, a simple harmonic oscillator with solution: \begin{align*} d^k(t) = \hat{g}_k \cos\bigl(\sqrt{\lambda_k}\,t\bigr) + \frac{\hat{h}_k}{\sqrt{\lambda_k}}\sin\bigl(\sqrt{\lambda_k}\,t\bigr), \end{align*} where $\hat{g}_k = (g, w_k)_{L^2}$ and $\hat{h}_k = (h, w_k)_{L^2}$. The full solution is: \begin{align*} u(x, t) = \sum_{k=1}^\infty \left[\hat{g}_k \cos\bigl(\sqrt{\lambda_k}\,t\bigr) + \frac{\hat{h}_k}{\sqrt{\lambda_k}}\sin\bigl(\sqrt{\lambda_k}\,t\bigr)\right] w_k(x). \end{align*} Compare with the [parabolic eigenfunction expansion](/pages/1074): the exponential decay $e^{-\lambda_k t}$ is replaced by oscillations $\cos(\sqrt{\lambda_k}\,t)$ and $\sin(\sqrt{\lambda_k}\,t)$. The amplitudes $\hat{g}_k$ and $\hat{h}_k/\sqrt{\lambda_k}$ do not decay with time — confirming the absence of smoothing. The solution oscillates forever without settling to equilibrium, and energy conservation reads: \begin{align*} E(t) = \frac{1}{2}\sum_{k=1}^\infty \left[\lambda_k \hat{g}_k^2 + \hat{h}_k^2\right] = E(0). \end{align*} [/example] [example: D'Alembert's Formula in One Dimension] On $\mathbb{R}$ with $g \in C^2(\mathbb{R})$, $h \in C^1(\mathbb{R})$, and $f = 0$, the wave equation $u_{tt} - u_{xx} = 0$ has the explicit solution: \begin{align*} u(x, t) = \frac{1}{2}\bigl[g(x + t) + g(x - t)\bigr] + \frac{1}{2}\int_{x-t}^{x+t} h(y)\,dy. \end{align*} This is **d'Alembert's formula**. The first term shows the initial displacement $g$ splitting into two waves traveling in opposite directions at speed $1$. The formula makes finite speed of propagation transparent: the value $u(x, t)$ depends only on $g$ and $h$ in the interval $[x - t, x + t]$, and changes in the initial data outside this interval have no effect. It also exhibits the absence of smoothing: if $g$ has a corner at $x_0$, then $u(\cdot, t)$ has corners at $x_0 + t$ and $x_0 - t$ for all $t > 0$. [/example] ## Problems [problem] **(Energy conservation for the wave equation.)** Let $u$ be a smooth solution of $u_{tt} - \Delta u = 0$ in $U \times (0, T]$ with $u = 0$ on $\partial U$. 1. Define $E(t) := \frac{1}{2}\int_U \bigl(u_t^2 + |\nabla u|^2\bigr)\,d\mathcal{L}^n$. Prove that $E'(t) = 0$ for all $t$. 2. Deduce that $\|u_t(t)\|_{L^2}^2 + \|\nabla u(t)\|_{L^2}^2 = \|h\|_{L^2}^2 + \|\nabla g\|_{L^2}^2$ for all $t$. 3. Explain why this implies that the wave equation has no analogue of the Poincaré-based exponential decay $\|u(t)\|_{L^2} \le e^{-t/C_P^2}\|g\|_{L^2}$ proved in the [parabolic energy problem](/pages/1074). [/problem] [solution] **Part 1.** Differentiate under the [integral](/page/Integral): \begin{align*} E'(t) = \int_U \bigl(u_t u_{tt} + \nabla u \cdot \nabla u_t\bigr)\,d\mathcal{L}^n. \end{align*} Use the PDE $u_{tt} = \Delta u$ to replace $u_{tt}$: \begin{align*} E'(t) = \int_U \bigl(u_t \Delta u + \nabla u \cdot \nabla u_t\bigr)\,d\mathcal{L}^n. \end{align*} Integrate $u_t \Delta u$ by parts (using $u = 0$ on $\partial U$, so $u_t = 0$ on $\partial U$): \begin{align*} \int_U u_t \Delta u\,d\mathcal{L}^n = -\int_U \nabla u_t \cdot \nabla u\,d\mathcal{L}^n. \end{align*} The two terms cancel: $E'(t) = -\int_U \nabla u_t \cdot \nabla u\,d\mathcal{L}^n + \int_U \nabla u \cdot \nabla u_t\,d\mathcal{L}^n = 0$. **Part 2.** Since $E'(t) = 0$, we have $E(t) = E(0)$ for all $t$. At $t = 0$: $E(0) = \frac{1}{2}(\|h\|_{L^2}^2 + \|\nabla g\|_{L^2}^2)$. The identity follows. **Part 3.** The parabolic exponential decay relies on the energy being monotonically decreasing — the dissipation term $-2\|\nabla u\|_{L^2}^2$ in the parabolic energy identity drives $\|u(t)\|_{L^2} \to 0$. For the wave equation, the energy is constant: there is no dissipation to exploit. The kinetic and potential energies oscillate but their sum never decreases, so the solution never decays to zero. In particular, the Poincaré inequality cannot be used to convert spatial gradient control into $L^2$ decay, because the gradient energy is not being "spent" — it is being traded back and forth with kinetic energy. [/solution] [problem] **(Finite speed of propagation.)** Let $u$ solve $u_{tt} - u_{xx} = 0$ on $\mathbb{R} \times (0, \infty)$ with $u(x, 0) = g(x)$ and $u_t(x, 0) = 0$, where $g$ is supported in $[-1, 1]$. 1. Use d'Alembert's formula to show that $u(x, t) = 0$ for $|x| > 1 + t$. 2. Show that the support of $u(\cdot, t)$ is contained in $[-(1+t), 1+t]$ for all $t \ge 0$. 3. Contrast this with the heat equation: if $v_t - v_{xx} = 0$ with $v(x, 0) = g(x)$, explain (using the heat kernel or the strong maximum principle) why $v(x, t) > 0$ for all $x \in \mathbb{R}$ and all $t > 0$, regardless of the support of $g$. [/problem] [solution] **Part 1.** With $h = 0$, d'Alembert gives $u(x, t) = \frac{1}{2}[g(x+t) + g(x-t)]$. If $|x| > 1 + t$, then $|x + t| > 1$ and $|x - t| > 1$ (since $|x \pm t| \ge |x| - t > 1$), so both $g(x+t) = 0$ and $g(x-t) = 0$, giving $u(x,t) = 0$. **Part 2.** The same argument: $u(x, t) \neq 0$ requires $g(x+t) \neq 0$ or $g(x-t) \neq 0$, which requires $|x+t| \le 1$ or $|x-t| \le 1$, i.e. $x \in [-(1+t), 1+t]$. **Part 3.** The heat kernel on $\mathbb{R}$ is $\Phi(x, t) = (4\pi t)^{-1/2} e^{-x^2/(4t)}$, which is strictly positive for all $x \in \mathbb{R}$ and $t > 0$. The solution $v(x, t) = (\Phi(\cdot, t) * g)(x) = \int_\mathbb{R} \Phi(x - y, t)\,g(y)\,dy$ is strictly positive whenever $g \ge 0$, $g \not\equiv 0$, because the integrand is non-negative and strictly positive on a [set](/page/Set) of positive measure. Alternatively, the strong [parabolic maximum principle](/pages/1074) applied to $-v$ (which satisfies $(-v)_t - (-v)_{xx} = 0 \le 0$ with $-v \le 0$ on the initial data) shows that $-v$ cannot attain its maximum $0$ at any interior point unless $v \equiv 0$. Since $g \not\equiv 0$, we have $v > 0$ everywhere for $t > 0$. This is the infinite speed of propagation: the parabolic smoothing kernel $\Phi(x, t)$ has Gaussian tails extending to $\pm\infty$, while the hyperbolic propagation is confined to the interval $[x - t, x + t]$. [/solution] ## References 1. L. C. Evans, *Partial Differential Equations*, 2nd ed., AMS (2010). Ch. 7, §7.2. 2. L. C. Evans, *Partial Differential Equations*, 2nd ed., AMS (2010). Ch. 2, §2.4 (wave equation in explicit form, d'Alembert). 3. J. Wloka, *Partial Differential Equations*, Cambridge University Press (1987). Ch. V. 4. R. Courant and D. Hilbert, *Methods of Mathematical Physics*, Vol. II (1962). Ch. VI. 5. F. John, *Partial Differential Equations*, 4th ed., Springer (1982).