The [second-order hyperbolic theory](/pages/1075) treats a single scalar equation $u_{tt} + Lu = f$ for one unknown [function](/page/Function) $u$. Many important physical systems — the equations of gas dynamics, electromagnetism, elasticity — naturally involve **several coupled unknowns** evolving simultaneously. These are described by first-order hyperbolic systems, where the unknown is a vector $u = (u^1, \ldots, u^m)$ and the spatial [derivatives](/page/Derivative) are coupled through matrix coefficients. {width=80%} The passage from scalar to system is not merely a matter of generality: scalar second-order equations can always be **reduced** to first-order systems (by introducing auxiliary variables for derivatives), and conversely, first-order systems often arise directly from conservation laws rather than from higher-order equations. The first-order formulation makes the wave-propagation structure explicit: the eigenvalues of the coefficient matrices are the **characteristic speeds** at which information travels, and the eigenvectors determine which combinations of the unknowns propagate along each characteristic. ## The General First-Order System [definition: First-Order Hyperbolic System] A **first-order system of PDEs** in $m$ unknowns is: \begin{align*} u_t + \sum_{k=1}^n A^k(x,t)\,u_{x_k} + B(x,t)\,u = f(x,t) \qquad \text{in } \mathbb{R}^n \times (0, T], \end{align*} with initial condition $u(\cdot, 0) = g$ in $\mathbb{R}^n$. Here $u: \mathbb{R}^n \times [0, T] \to \mathbb{R}^m$ is the unknown vector, $A^k(x,t)$ are $m \times m$ matrices for $k = 1, \ldots, n$, $B(x,t)$ is an $m \times m$ matrix of lower-order coefficients, and $f$ is a vector-valued source. [/definition] The system is written more compactly as $u_t + \sum_k A^k u_{x_k} + Bu = f$. The term "first-order" refers to the fact that only first derivatives of $u$ appear (both in $t$ and in $x$), in contrast to the [second-order scalar equation](/pages/1075) $u_{tt} + Lu = f$ where second derivatives of $u$ occur. The character of the system — whether it exhibits wave-like propagation or diffusive behaviour — is determined entirely by the coefficient matrices $A^k$. The key concept is: [definition: Symmetric Hyperbolicity] The system $u_t + \sum_k A^k u_{x_k} + Bu = f$ is **symmetric hyperbolic** if the matrices $A^k(x,t)$ are all **symmetric** ($A^k = (A^k)^\top$) for every $(x,t)$. [/definition] Symmetry of the coefficient matrices is the first-order analogue of the uniform ellipticity condition $\sum a_{ij}\xi_i\xi_j \ge \theta|\xi|^2$ that appears throughout the [elliptic](/pages/1033), [parabolic](/pages/1074), and [hyperbolic](/pages/1075) scalar theories. Just as ellipticity ensures coercivity of the bilinear form, symmetry of the $A^k$ ensures that the natural energy $\frac{1}{2}\|u(t)\|_{L^2}^2$ satisfies a useful energy identity — the antisymmetric part would destroy this. [remark: Why Symmetry?] The reason symmetry is the right condition becomes clear when one derives the energy estimate (see below). Testing the equation with $u$ and integrating by parts produces terms of the form $\int u \cdot A^k u_{x_k}\,dx$. [Integration](/page/Integral) by parts gives: \begin{align*} \int_{\mathbb{R}^n} u \cdot A^k u_{x_k}\,dx = -\int_{\mathbb{R}^n} u_{x_k} \cdot A^k u\,dx - \int_{\mathbb{R}^n} u \cdot A^k_{x_k} u\,dx. \end{align*} If $A^k$ is symmetric, the first term on the right equals $-\int u \cdot A^k u_{x_k}\,dx$, so the two sides combine to give $2\int u \cdot A^k u_{x_k}\,dx = -\int u \cdot A^k_{x_k} u\,dx$, which is bounded by lower-order terms. Without symmetry, this cancellation fails. [/remark] ## Characteristics and Propagation Speeds The physical content of a hyperbolic system is encoded in its **characteristics** — the directions along which information propagates. For a symmetric hyperbolic system, these are determined by the eigenvalues of the coefficient matrices. [definition: Characteristic Speeds] For a direction $\xi \in \mathbb{R}^n \setminus \{0\}$, the **symbol** of the system at $(x,t)$ in direction $\xi$ is the $m \times m$ matrix: \begin{align*} A(x,t,\xi) := \sum_{k=1}^n \xi_k\,A^k(x,t). \end{align*} Since each $A^k$ is symmetric, $A(x,t,\xi)$ is symmetric for every $\xi$ and hence has $m$ real eigenvalues $\lambda_1(x,t,\xi) \le \cdots \le \lambda_m(x,t,\xi)$. These are the **characteristic speeds** of the system in direction $\xi$. [/definition] {width=75%} Each eigenvalue $\lambda_k$ represents a speed at which a particular "mode" of the solution propagates in direction $\xi$. The corresponding eigenvector determines which linear combination of the components $u^1, \ldots, u^m$ travels at that speed. The extreme eigenvalues $\lambda_1$ and $\lambda_m$ bound the speed of propagation: no information travels faster than $\max_{|\xi|=1}|\lambda_m(x,t,\xi)|$. [remark: Reduction of Scalar Equations to Systems] The [second-order wave equation](/pages/1075) $u_{tt} - u_{xx} = 0$ (in one spatial dimension) can be written as a first-order symmetric hyperbolic system by introducing $v := u_t$ and $w := u_x$. The equation $u_{tt} = u_{xx}$ becomes $v_t = w_x$, and the compatibility condition $u_{tx} = u_{xt}$ gives $w_t = v_x$. In matrix form: \begin{align*} \begin{pmatrix} v \\ w \end{pmatrix}_t - \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} v \\ w \end{pmatrix}_x = \begin{pmatrix} 0 \\ 0 \end{pmatrix}. \end{align*} The coefficient matrix $A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ is symmetric with eigenvalues $\lambda = \pm 1$ — the two wave speeds — and eigenvectors $(1, \pm 1)^\top$. The combinations $v + w = u_t + u_x$ and $v - w = u_t - u_x$ are the **Riemann invariants**, each propagating along one family of characteristics. {width=80%} More generally, any [second-order hyperbolic equation](/pages/1075) $u_{tt} + Lu = f$ in $n$ spatial dimensions can be converted to a first-order symmetric system in $m = n + 1$ unknowns $u = (u_t, u_{x_1}, \ldots, u_{x_n})$. This is one motivation for developing the first-order theory. [/remark] ## Energy Estimates The fundamental tool for symmetric hyperbolic systems is, as in all the preceding theories, an energy estimate. The symmetry of the $A^k$ makes this particularly clean. [theorem: Energy Estimate for Symmetric Hyperbolic Systems] Let $u$ be a smooth solution of the symmetric hyperbolic system $u_t + \sum_k A^k u_{x_k} + Bu = f$ in $\mathbb{R}^n \times (0, T]$ with $u(\cdot, 0) = g$. Assume $A^k, B \in C^1$ with bounded derivatives. Then there exists a constant $C > 0$, depending on $T$, $\|B\|_{L^\infty}$, and $\|\partial_{x_k} A^k\|_{L^\infty}$, such that: \begin{align*} \max_{0 \le t \le T} \|u(t)\|_{L^2(\mathbb{R}^n)}^2 \le C\left(\|g\|_{L^2(\mathbb{R}^n)}^2 + \int_0^T \|f(t)\|_{L^2(\mathbb{R}^n)}^2\,dt\right). \end{align*} [/theorem] [proof] **Step 1: Energy computation.** Define $e(t) := \frac{1}{2}\int_{\mathbb{R}^n} |u(x,t)|^2\,d\mathcal{L}^n$. Differentiate: \begin{align*} e'(t) = \int_{\mathbb{R}^n} u \cdot u_t\,d\mathcal{L}^n = \int_{\mathbb{R}^n} u \cdot \left(f - \sum_k A^k u_{x_k} - Bu\right) d\mathcal{L}^n. \end{align*} **Step 2: [Integration by parts](/theorems/210).** The key term is $\int u \cdot A^k u_{x_k}\,d\mathcal{L}^n$. Since $A^k$ is symmetric, integration by parts (with the [boundary](/page/Boundary) terms vanishing since $u$ is defined on all of $\mathbb{R}^n$ and decays at infinity) gives: \begin{align*} \int_{\mathbb{R}^n} u \cdot A^k u_{x_k}\,d\mathcal{L}^n = -\frac{1}{2}\int_{\mathbb{R}^n} u \cdot (\partial_{x_k} A^k)\,u\,d\mathcal{L}^n. \end{align*} This is the step where symmetry is essential: it converts a first-order derivative term into a zeroth-order term. **Step 3: Gronwall.** Combining: \begin{align*} e'(t) \le \frac{1}{2}\sum_k \|\partial_{x_k} A^k\|_{L^\infty}\int |u|^2\,d\mathcal{L}^n + \|B\|_{L^\infty}\int |u|^2\,d\mathcal{L}^n + \int |u||f|\,d\mathcal{L}^n. \end{align*} By Cauchy-Schwarz and Young's inequality on the last term: $e'(t) \le C_0\,e(t) + \frac{1}{2}\|f(t)\|_{L^2}^2$, where $C_0$ depends on the coefficient bounds. Gronwall's inequality gives the result. [/proof] [remark: Comparison With Second-Order Energy Estimates] The energy estimate for systems controls $\|u(t)\|_{L^2}^2$ — the $L^2$ norm of the solution vector itself. For the [second-order hyperbolic equation](/pages/1075), the energy involves both $\|u_t\|_{L^2}^2$ and $\|\nabla u\|_{L^2}^2$ (kinetic and potential energy). These are equivalent formulations: when the second-order equation $u_{tt} - \Delta u = 0$ is written as the first-order system with $u = (u_t, u_{x_1}, \ldots, u_{x_n})$, the $L^2$ norm $\|u\|_{L^2}^2 = \|u_t\|_{L^2}^2 + \|\nabla u\|_{L^2}^2$ recovers the total energy. [/remark] As in all preceding theories, the energy estimate immediately gives **uniqueness**: if $u_1$ and $u_2$ solve the same system with the same data, then $w = u_1 - u_2$ solves the homogeneous system with $g = 0$ and $f = 0$, so $\|w(t)\|_{L^2} = 0$ for all $t$. ## Existence of Solutions The existence theory for symmetric hyperbolic systems differs from the Galerkin approach used for [parabolic](/pages/1074) and [second-order hyperbolic](/pages/1075) equations. The natural method is **Friedrichs' mollification**: regularise the coefficients and data, solve the resulting smooth problem, and pass to the limit using the energy estimate. [motivation] ### The Strategy 1. **Mollify.** Replace $A^k$, $B$, $f$, $g$ by smooth approximations $A^{k,\varepsilon}$, $B^\varepsilon$, $f^\varepsilon$, $g^\varepsilon$ via [convolution](/page/Convolution) with a [standard mollifier](/page/Standard%20Mollifier). 2. **Solve the smooth system.** For smooth, bounded coefficients, the system $u^\varepsilon_t + \sum_k A^{k,\varepsilon} u^\varepsilon_{x_k} + B^\varepsilon u^\varepsilon = f^\varepsilon$ with $u^\varepsilon(\cdot, 0) = g^\varepsilon$ can be solved by the [method of characteristics](/page/Method%20of%20Characteristics) (locally) or by Picard iteration in appropriate function spaces. 3. **Uniform energy estimates.** Since each $A^{k,\varepsilon}$ is symmetric, the energy estimate applies uniformly in $\varepsilon$: $\|u^\varepsilon(t)\|_{L^2}^2 \le C(\|g^\varepsilon\|_{L^2}^2 + \int_0^T \|f^\varepsilon\|_{L^2}^2\,dt)$, with $C$ independent of $\varepsilon$. 4. **Passage to limit.** The uniform bounds give [weak convergence](/page/Weak%20Convergence) of a subsequence $u^{\varepsilon_j} \rightharpoonup u$ in $L^2$. The [limit](/page/Limit) $u$ is verified to be a weak solution of the original system. [/motivation] [theorem: Existence for Symmetric Hyperbolic Systems] Let $A^k$ be symmetric $m \times m$ matrices with $A^k, B \in L^\infty(\mathbb{R}^n \times [0, T])$, with $\partial_{x_k} A^k \in L^\infty$. Let $f \in L^2(0, T; L^2(\mathbb{R}^n; \mathbb{R}^m))$ and $g \in L^2(\mathbb{R}^n; \mathbb{R}^m)$. Then there exists a unique weak solution $u \in L^2(0, T; L^2(\mathbb{R}^n; \mathbb{R}^m))$ of: \begin{align*} \begin{cases} u_t + \sum_{k=1}^n A^k u_{x_k} + Bu = f & \text{in } \mathbb{R}^n \times (0, T], \\ u(\cdot, 0) = g & \text{in } \mathbb{R}^n, \end{cases} \end{align*} with $u \in C([0, T]; L^2(\mathbb{R}^n; \mathbb{R}^m))$. Moreover, $u$ satisfies the energy estimate. [/theorem] ## Finite Speed of Propagation As for the [scalar wave equation](/pages/1075), symmetric hyperbolic systems propagate disturbances at finite speed. The maximum speed is determined by the largest eigenvalue of the symbol $A(x,t,\xi)$ over all directions $\xi$. [theorem: Finite Speed of Propagation for Systems] Let $u$ be the solution of the symmetric hyperbolic system with $f = 0$ and initial data $g$ supported in a ball $B(x_0, r) \subset \mathbb{R}^n$. Define the **maximum characteristic speed**: \begin{align*} c := \sup_{(x,t) \in \mathbb{R}^n \times [0,T]} \,\sup_{|\xi| = 1} \,\max_{1 \le j \le m} |\lambda_j(x,t,\xi)|. \end{align*} Then $u(\cdot, t)$ is supported in $B(x_0, r + ct)$ for all $t \in [0, T]$. [/theorem] The proof uses a domain-of-dependence argument: define the energy in the truncated cone $K_\tau := \{(x, t) : |x - x_0| < r + c(T - t),\; 0 < t < \tau\}$ and show that the energy flux through the lateral boundary is non-positive (this is where the bound on the eigenvalues enters). Since the initial data vanishes outside $B(x_0, r)$, the energy outside the cone is initially zero and remains zero. [remark: Domain of Dependence] The **domain of dependence** of a point $(x_0, t_0)$ is the [set](/page/Set) of initial points $(y, 0)$ that can influence $u(x_0, t_0)$. For a symmetric hyperbolic system with maximum speed $c$, this domain is contained in $B(x_0, ct_0)$. Conversely, the **domain of influence** of the ball $B(x_0, r)$ at time $t$ is $B(x_0, r + ct)$. When the system has $m$ distinct eigenvalues $\lambda_1 < \cdots < \lambda_m$ (the **strictly hyperbolic** case), each eigenvalue defines a separate family of characteristics, and the domain of dependence is bounded by the extreme characteristics $\lambda_1$ and $\lambda_m$. The solution can be decomposed into $m$ "waves," each propagating along its own characteristic family. [/remark] ## Higher Regularity The regularity theory for symmetric hyperbolic systems is analogous to the [second-order hyperbolic theory](/pages/1075): regularity is preserved, not gained. The key tool is **differentiating the equation**: if $u$ solves the system, then $\partial^\alpha u$ (for any multi-index $\alpha$) satisfies a system of the same type, with modified lower-order terms. The energy estimate applied to the differentiated system controls higher derivatives of $u$ in terms of higher derivatives of the data. [theorem: Higher Regularity for Symmetric Hyperbolic Systems] Assume the coefficients $A^k, B$ are $C^\infty$ with bounded derivatives of all orders. If $g \in H^s(\mathbb{R}^n; \mathbb{R}^m)$ and $f \in L^2(0, T; H^s(\mathbb{R}^n; \mathbb{R}^m))$ for some integer $s \ge 0$, then the unique solution satisfies: \begin{align*} u \in C^k([0, T]; H^{s-k}(\mathbb{R}^n; \mathbb{R}^m)) \quad \text{for } k = 0, 1, \ldots, s. \end{align*} In particular, $u \in C([0, T]; H^s(\mathbb{R}^n))$ and each time derivative costs one spatial derivative, just as in the [second-order theory](/pages/1075). [/theorem] The proof applies the energy estimate to the system satisfied by $D^\alpha u$ for $|\alpha| \le s$. Differentiating $u_t + \sum_k A^k u_{x_k} + Bu = f$ with respect to $x$ introduces commutator terms $[D^\alpha, A^k]u_{x_k}$, which involve at most $|\alpha|$ derivatives of $u$ (one fewer than the leading term) and are controlled by induction. As with the [second-order theory](/pages/1075), there is no smoothing: $g \in H^s$ gives $u(\cdot, t) \in H^s$ for all $t$, but never $H^{s+1}$. Singularities propagate along characteristics at finite speed. ## Examples [example: The Wave Equation as a System] The one-dimensional [wave equation](/page/Wave%20Equation) $u_{tt} - c^2 u_{xx} = 0$ with wave speed $c > 0$ reduces to the system: \begin{align*} \begin{pmatrix} v \\ w \end{pmatrix}_t - c\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} v \\ w \end{pmatrix}_x = 0, \end{align*} where $v = u_t$ and $w = cu_x$. The matrix $A = c\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ has eigenvalues $\pm c$ with eigenvectors $(1, \pm 1)^\top$. The **Riemann invariants** $r_\pm = v \pm w = u_t \pm cu_x$ satisfy the decoupled transport equations $(r_+)_t - c(r_+)_x = 0$ and $(r_-)_t + c(r_-)_x = 0$. That is, $r_+$ propagates to the right at speed $c$ and $r_-$ propagates to the left at speed $c$. This is exactly the content of [d'Alembert's formula](/pages/1075): $u(x,t) = f(x+ct) + g(x-ct)$. [/example] [example: Maxwell's Equations] Maxwell's equations for the electric field $E = (E_1, E_2, E_3)$ and magnetic field $B = (B_1, B_2, B_3)$ in vacuum are: \begin{align*} E_t = c\,\nabla \times B, \qquad B_t = -c\,\nabla \times E, \end{align*} where $c$ is the speed of light (together with the constraints $\nabla \cdot E = 0$, $\nabla \cdot B = 0$). Writing $u = (E, B)^\top \in \mathbb{R}^6$, this is a $6 \times 6$ symmetric hyperbolic system $u_t + \sum_{k=1}^3 A^k u_{x_k} = 0$, where the matrices $A^k$ encode the curl operators. Each $A^k$ is symmetric and has eigenvalues $0, 0, \pm c, \pm c$ (the double zero eigenvalues correspond to the constraint modes, and $\pm c$ are the light-speed modes). The characteristic speeds are $\pm c$ and $0$: electromagnetic waves propagate at exactly the speed of light. [/example] [example: Linearised Euler Equations] The equations governing small perturbations of a uniform gas at rest are: \begin{align*} \rho_t + \bar{\rho}\,\nabla \cdot v = 0, \qquad v_t + \frac{\bar{c}^2}{\bar{\rho}}\nabla \rho = 0, \end{align*} where $\rho$ is the density perturbation, $v = (v_1, \ldots, v_n)$ is the velocity perturbation, $\bar{\rho}$ is the background density, and $\bar{c}$ is the sound speed. This is a symmetric hyperbolic system for $u = (\rho, v_1, \ldots, v_n)^\top \in \mathbb{R}^{n+1}$. In one spatial dimension, the coefficient matrix is $A = \begin{pmatrix} 0 & \bar{\rho} \\ \bar{c}^2/\bar{\rho} & 0 \end{pmatrix}$ with eigenvalues $\pm \bar{c}$ — the sound speed. The two characteristics represent sound waves traveling left and right. [/example] ## Problems [problem] **(Riemann invariants for the wave equation.)** Consider the system $u_t + Au_x = 0$ where $u = (v, w)^\top$ and $A = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$. 1. Find the eigenvalues and eigenvectors of $A$. 2. Define new variables $r_+ := v - w$ and $r_- := v + w$. Show that $r_+$ and $r_-$ each satisfy a scalar transport equation. 3. Solve the system explicitly given $v(x, 0) = \varphi(x)$ and $w(x, 0) = \psi(x)$. 4. Verify that the solution exhibits finite speed of propagation with speed $1$. [/problem] [solution] **Part 1.** The characteristic equation is $\det(A - \lambda I) = \lambda^2 - 1 = 0$, giving $\lambda_1 = -1$ and $\lambda_2 = 1$. The eigenvectors are $r_1 = (1, 1)^\top$ (for $\lambda_1 = -1$) and $r_2 = (1, -1)^\top$ (for $\lambda_2 = 1$). **Part 2.** Compute: $(r_+)_t = v_t - w_t = (w_x) - (v_x) = -(r_+)_x + 2w_x$. More directly, from the system $v_t + w_x = 0$ and $w_t + v_x = 0$, we get $(v - w)_t = -(w_x - v_x) = (v - w)_x$, i.e., $(r_+)_t - (r_+)_x = 0$. Similarly, $(v + w)_t = -(w_x + v_x) = -(v + w)_x$, i.e., $(r_-)_t + (r_-)_x = 0$. So $r_+$ satisfies the transport equation with speed $+1$ (right-moving) and $r_-$ with speed $-1$ (left-moving). **Part 3.** The transport equations give $r_+(x, t) = r_+(x + t, 0) = \varphi(x + t) - \psi(x + t)$ and $r_-(x, t) = r_-(x - t, 0) = \varphi(x - t) + \psi(x - t)$. Recovering $v$ and $w$: \begin{align*} v(x, t) &= \frac{1}{2}(r_+ + r_-) = \frac{1}{2}[\varphi(x+t) - \psi(x+t) + \varphi(x-t) + \psi(x-t)], \\ w(x, t) &= \frac{1}{2}(r_- - r_+) = \frac{1}{2}[-\varphi(x+t) + \psi(x+t) + \varphi(x-t) + \psi(x-t)]. \end{align*} **Part 4.** The value $(v(x,t), w(x,t))$ depends only on $\varphi$ and $\psi$ evaluated at $x + t$ and $x - t$, i.e., on the initial data in the interval $[x - t, x + t]$. If $\varphi$ and $\psi$ are supported in $[-R, R]$, then $u(\cdot, t)$ is supported in $[-(R + t), R + t]$ — the speed of propagation is exactly $1$, matching the eigenvalues $\lambda = \pm 1$. [/solution] [problem] **(Energy estimate for a $2 \times 2$ system.)** Let $u = (u^1, u^2)^\top$ solve the symmetric system $u_t + Au_x = 0$ on $\mathbb{R} \times (0, T]$ where $A = \begin{pmatrix} a & b \\ b & d \end{pmatrix}$ is a constant symmetric matrix. 1. Define $E(t) := \frac{1}{2}\int_\mathbb{R} |u(x,t)|^2\,dx$. Show that $E(t) = E(0)$ for all $t$. 2. Explain why the result fails if $A$ is not symmetric. 3. If instead $u_t + Au_x + Bu = 0$ with $B$ a constant matrix, show that $E(t) \le e^{2\|B\|t}E(0)$ (growth, but no blow-up in finite time). [/problem] [solution] **Part 1.** Differentiate: $E'(t) = \int_\mathbb{R} u \cdot u_t\,dx = -\int_\mathbb{R} u \cdot Au_x\,dx$. Integrate by parts (with vanishing boundary terms): $\int u \cdot Au_x\,dx = -\int u_x \cdot Au\,dx$ (since $A$ is constant, no $A_x$ term). Since $A$ is symmetric, $u_x \cdot Au = u \cdot Au_x$, so $\int u \cdot Au_x\,dx = -\int u \cdot Au_x\,dx$, giving $\int u \cdot Au_x\,dx = 0$. Hence $E'(t) = 0$. **Part 2.** If $A$ is not symmetric, then $u_x \cdot Au \neq u \cdot Au_x$ in general, so the cancellation fails. Write $A = S + K$ where $S = \frac{1}{2}(A + A^\top)$ and $K = \frac{1}{2}(A - A^\top)$. The symmetric part $S$ contributes zero (by Part 1), but the antisymmetric part $K$ contributes $\int u \cdot Ku_x\,dx$, which after integration by parts gives $-\int u_x \cdot Ku\,dx = +\int u \cdot Ku_x\,dx$ (since $K^\top = -K$). So $2\int u \cdot Ku_x\,dx = 0$ — actually the antisymmetric part also contributes zero for constant $K$! The issue arises only when the coefficients depend on $x$: then $\partial_x K$ terms appear and do not cancel. **Part 3.** With the lower-order term: $E'(t) = -\int u \cdot Bu\,dx \le \|B\|\int |u|^2\,dx = 2\|B\|\,E(t)$. Gronwall's inequality gives $E(t) \le e^{2\|B\|t}E(0)$. [/solution] ## References 1. L. C. Evans, *Partial Differential Equations*, 2nd ed., AMS (2010). Ch. 7, §7.3. 2. K. O. Friedrichs, "Symmetric hyperbolic linear differential equations," *Comm. Pure Appl. Math.* **7** (1954), 345–392. 3. R. Courant and D. Hilbert, *Methods of Mathematical Physics*, Vol. II (1962). Ch. VI. 4. F. John, *Partial Differential Equations*, 4th ed., Springer (1982). Ch. 5. 5. J. Rauch, *Hyperbolic Partial Differential Equations and Geometric Optics*, AMS (2012).