Galerkin Existence and Finite Propagation Speed for Second-Order Hyperbolic Equations

Intuition Beginner

A wave equation describes a quantity that oscillates rather than settles: a plucked string, a drumhead, a sound pulse in air, a tremor through rock. The companion unit on the wave equation handed you one such equation on an infinite line, with constant wave speed, and one explicit formula, the splitting of a bump into a left-mover and a right-mover. But most real vibration happens in a bounded region, through a material whose stiffness varies from place to place and may even change in time. There is no tidy splitting formula for that. The sharp question returns: does a solution even exist, and is it the only one?

This unit answers yes by building the solution out of simple pieces instead of guessing a formula, exactly as the parabolic companion did for diffusion. Pick a handful of standard shapes a function on the region can have, the lowest vibration modes, and look for the best blend of just those few shapes. Each shape carries a strength that changes in time, and feeding the blend into the wave rule turns the partial differential equation into an ordinary system of rate equations for those strengths, the kind of system that always has a solution. Then add more shapes, and more, and watch the approximations settle toward an honest solution.

Why should adding shapes settle down? Because a vibrating system has a total energy, part stored in motion and part stored in stretch, and that total stays put. It does not leak away the way heat does; it only sloshes back and forth between the two stores, apart from whatever a driving force feeds in. That fixed energy budget turns into a uniform numerical ceiling on every approximation at once, no matter how many shapes were used, and a uniformly capped family cannot run off to infinity.

Here is the sharp difference from the diffusion companion. Diffusion drains energy, so its solutions only get smoother. A wave conserves energy, so a kink stays a kink: there is no smoothing, no regularity gained for free. Conservation also carries a new fact diffusion lacks. A wiggle started in one small region cannot be felt arbitrarily far away right away; it can spread only at a bounded speed, so after a given time it lives inside a cone. That bounded reach, finite propagation speed, is the second main result of this unit, and it holds even when the material varies.

Visual Beginner

Two pictures carry this unit: an energy that swaps between two stores without leaking, and a cone that bounds how far a disturbance can reach.

Read the left panel first. A vibrating system holds energy in two forms: the energy of motion, set by how fast each piece is moving, and the energy of stretch, set by how bent or strained the shape is. As the system vibrates, energy pours from one tank into the other and back, but the sealed lid means the total never falls on its own. Only an outside driving force, the small inlet, can change the total, and only by a controlled amount. This sealed-total picture is the conservation law, and it is the single energy ceiling that holds every shape-by-shape approximation at once.

The right panel is the cone. Start a disturbance inside a small interval at time zero. A signal can travel only so fast, so by a later time the disturbance can have reached only the points within that travel distance of where it began. Plotting space across and time up, the reachable region is a triangle widening as time grows, with slanted sides whose steepness is set by the fastest signal speed.

A point outside the triangle has not yet been touched: the medium there is still exactly at rest. That bounded reach is finite propagation speed, and it survives even when the material, and hence the local signal speed, varies from place to place, as long as the cone is drawn with a slope at least the largest local speed.

The contrast with the diffusion companion is the whole point. There the right-hand picture was a single energy curve sloping downward, energy leaking away and the solution smoothing. Here the energy curve is flat and the reach is bounded by a cone. Conservation in place of dissipation, a cone in place of an infinite instantaneous reach.

Worked example Beginner

We run the shape-by-shape recipe by hand on the simplest bounded vibration and watch the energy stay fixed. Take a string from zero to $\pi$, pinned at zero displacement at both ends, governed by the wave rule "acceleration equals curvature", which is $u_{tt}=u_{xx}$. Start it from the shape $u(x,0)=3\sin x$ released from rest, so the initial velocity is zero.

Step 1. Choose the shapes. The natural shapes for a string pinned at both ends are the standing waves $\sin x$, $\sin 2x$, $\sin 3x$, and so on. Each keeps its shape under the wave rule and only changes in strength.

Step 2. Write the blend. Look for $u(x,t)=a_1(t)\sin x$, a single shape since the start is a single shape. The starting strength is $a_1(0)=3$, and the starting rate of change is $a_1^{\prime }(0)=0$ because the string is released from rest.

Step 3. Turn the rule into a rate equation. The curvature of $\sin x$ is $-1\cdot \sin x$. Matching acceleration to curvature gives $a_1^{\prime \prime }(t)=-1\cdot a_1(t)$. This is the equation of a mass on a spring: the strength oscillates rather than decays.

Step 4. Solve it. The solution with strength $3$ and zero starting rate is $a_1(t)=3\cos t$. So $u(x,t)=3\cos t\sin x$. The whole string swings up and down in unison, a standing wave, returning to its start every time $t$ advances by $2\pi$.

Step 5. Watch the energy stay fixed. The energy of motion is set by the velocity $u_t=-3\sin t\sin x$, and the energy of stretch is set by the slope $u_x=3\cos t\cos x$. Adding the totals of $u_t^2$ and $u_x^2$ across the string and using that $\sin x$ and $\cos x$ each total $\frac{\pi }{2}$ in square gives $\frac{\pi }{2}(9{\sin }^{2}t+9{\cos }^{2}t)=\frac{\pi }{2}\cdot 9$. The ${\sin }^{2}t$ and ${\cos }^{2}t$ add to one at every instant, so the total is the same constant $\frac{9\pi }{2}$ for all time.

What this tells us: restricting to one shape turned the partial differential equation into a single spring equation we solved exactly, and the total energy, the sum of the motion part and the stretch part, never changed; it only poured back and forth between the two as ${\sin }^{2}t$ and ${\cos }^{2}t$ traded places. With no driving force, the energy ceiling is just the starting value, and every approximation stays under it forever. That fixed energy is the engine that, in the general case, keeps the shape-by-shape approximations from blowing up as more shapes are added, and the absence of any decay is exactly why a wave keeps its wiggles.

Check your understanding Beginner

Formal definition Intermediate+

Throughout, $\Omega \subseteq {\mathbb{R}}^n$ is open and bounded, $T>0$ is fixed, and the spatial operator at each time is the symmetric divergence-form second-order operator of 02.16.04, $$L(t)u=-\sum _{i,j=1}^n{\partial }_i(a^{ij}(x,t){\partial }_{j}u)+c(x,t)u,$$ with coefficients $a^{ij},c\in L^{\infty }(\Omega \times (0,T))$, $a^{ij}=a^{ji}$, $c\ge 0$, the time-derivatives ${\partial }_t{a}^{ij}\in L^{\infty }(\Omega \times (0,T))$, and uniform ellipticity ${\sum }_{ij}{a}^{ij}(x,t){\xi }_i{\xi }_j\ge \theta |\xi {|}^2$ for a.e. $(x,t)$ and all $\xi \in {\mathbb{R}}^n$, with $\theta >0$. The symmetry $a^{ij}=a^{ji}$ and the absence of first-order drift make $L(t)$ formally self-adjoint, the natural setting for a conserved energy. The time-dependent bilinear form is $$B[u,v;t]={\int }_{\Omega }(\sum _{ij}{a}^{ij}(\cdot ,t){\partial }_{j}u{\partial }_{i}v+c(\cdot ,t)uv)dx,$$ bounded on $H_0^1(\Omega )$ with constant $\Lambda$ uniformly in $t$, symmetric in $(u,v)$, and Gårding-coercive, $B[u,u;t]\ge \beta \Vert u{\Vert }_V^2-\gamma \Vert u{\Vert }_H^2$ uniformly in $t$, by the estimates of 02.16.04.

Definition (Gelfand triple and solution space). Let $V=H_0^1(\Omega )$ and $H=L^2(\Omega )$, with $V\hookrightarrow H$ continuous and dense, and form the Gelfand triple $V\hookrightarrow H\cong H^{\ast }\hookrightarrow V^{\ast }$ exactly as in 02.18.01, so $V^{\ast }=H^{-1}(\Omega )$ and the duality pairing $⟨\cdot ,\cdot ⟩:V^{\ast }\times V\to \mathbb{R}$ extends the inner product of $H$. The Bochner spaces $L^p(0,T;X)$ are those of 02.18.01. The second-order solution space for the hyperbolic problem is $$\mathcal{W}(0,T)=\{u\in L^2(0,T;V):u^{\prime }\in L^2(0,T;H),u^{\prime \prime }\in L^2(0,T;V^{\ast })\},$$ where $u^{\prime }$ and $u^{\prime \prime }$ are the weak (distributional) time-derivatives valued in $H$ and $V^{\ast }$ respectively. Every $u\in \mathcal{W}(0,T)$ has, after modification on a null set, $u\in C([0,T];H)$ and $u^{\prime }\in C([0,T];V^{\ast })$; under the sharper regularity of the existence theorem below one gains $u\in C([0,T];V)$ and $u^{\prime }\in C([0,T];H)$, which is what gives the two initial conditions a meaning in $V$ and $H$.

Definition (weak solution of the hyperbolic problem). Given $f\in L^2(0,T;H)$, $g\in V$, and $h\in H$, a function $u$ with $u\in L^{\infty }(0,T;V)$, $u^{\prime }\in L^{\infty }(0,T;H)$, $u^{\prime \prime }\in L^2(0,T;V^{\ast })$ is a weak solution of the initial/boundary-value problem $$u_{tt}+L(t)u=f\text{in }\Omega \times (0,T],u=0\text{on }\partial \Omega \times (0,T],u(\cdot ,0)=g,u_t(\cdot ,0)=h,$$ if $u(0)=g$ in $V$, $u^{\prime }(0)=h$ in $H$, and $$⟨u^{\prime \prime }(t),v⟩+B[u(t),v;t]=(f(t),v{)}_H\text{for all }v\in V\text{ and a.e. }t\in (0,T).$$ The equation is an identity in $V^{\ast }$ at a.e. time; $u^{\prime \prime }(t)\in V^{\ast }$ because $L(t)u(t)\in V^{\ast }$ for $u(t)\in V$. Two pieces of initial data are required, one for position in $V$ and one for velocity in $H$, the hyperbolic analogue of needing both $g$ and $h$ for the d'Alembert problem of 02.13.04.

Definition (Galerkin approximation). Let $\{w_k{\}}_{k\ge 1}\subseteq V$ be the orthonormal-in-$H$, orthogonal-in-$V$ Dirichlet eigenbasis $-\Delta w_k={\lambda }_k{w}_k$ of 02.16.04, complete in both $H$ and $V$. The $m$-th Galerkin approximation is $$u_m(t)=\sum _{k=1}^m{d}_m^k(t)w_k\in V_m:=\mathrm{span}\{w_1,\dots ,w_m\},$$ whose coefficient vector $d_m(t)$ solves the second-order finite-dimensional ODE system $$(u_m^{\prime \prime }(t),w_k{)}_H+B[u_m(t),w_k;t]=(f(t),w_k{)}_H(k=1,\dots ,m),$$ $$d_m^k(0)=(g,w_k{)}_H,(d_m^k{)}^{\prime }(0)=(h,w_k{)}_H.$$

Counterexamples to common slips Intermediate+

• Conservation, not dissipation. For the hyperbolic problem the natural energy $E_m(t)=\frac{1}{2}\Vert u_m^{\prime }(t){\Vert }_H^2+\frac{1}{2}B[u_m(t),u_m(t);t]$ is not monotone; its derivative is controlled but not sign-definite, in contrast to the parabolic energy of 02.18.01 that decays. Treating the hyperbolic energy as dissipative and expecting a smoothing gain is the central error: there is no parabolic regularization, so $u(t)$ is no smoother than $g$ and $u_t(t)$ no smoother than $h$.

• The velocity lives in $H$, the acceleration in $V^$.\ast Thefirsttime-derivative$u'$ofaweakhyperbolicsolutionliesin$L^\infty(0,T;H)$,onedegreebetterthantheparabolic$u' \in L^2(0,T;V^)$,becausetestingwith$u_m'$yieldscontrolof$|u_m'|_H$.Thesecondderivative$u''$dropsbackto$V^$.Demanding$u'' \in L^2(0,T;H)$ is a genuine regularity gain requiring more of the data, not part of the definition.

• Two initial conditions, not one. The equation is second-order in time, so $u(0)=g$ alone underdetermines the solution; the velocity $u^{\prime }(0)=h$ is independent data. A scheme that initializes only the position, copying the parabolic template, solves the wrong problem.

• Finite propagation speed needs the right cone slope. The local energy on a backward cone is nonincreasing only when the cone's lateral slope is at least the largest local characteristic speed $\sqrt{\Vert a^{ij}{\Vert }_{L^{\infty }}/\theta }$. A cone drawn too steeply (slope below the wave speed) can leak energy through its lateral boundary, and the monotonicity fails; the propagation speed is set by the coefficients, not chosen freely.

Key theorem with proof Intermediate+

Theorem (Galerkin existence, uniqueness, and energy estimate). Let $L(t)$ be symmetric, uniformly elliptic with $L^{\infty }$ coefficients and ${\partial }_t{a}^{ij}\in L^{\infty }$ on $\Omega \times (0,T)$, so $B[\cdot ,\cdot ;t]$ is bounded with constant $\Lambda$, symmetric, and Gårding-coercive. Then for every $f\in L^2(0,T;H)$, $g\in V$, and $h\in H$ there is a unique weak solution $u$ of the hyperbolic problem with $$u\in L^{\infty }(0,T;V),u^{\prime }\in L^{\infty }(0,T;H),u^{\prime \prime }\in L^2(0,T;V^{\ast }),$$ and it obeys the a priori estimate $$\Vert u^{\prime }{\Vert }_{L^{\infty }(0,T;H)}+\Vert u{\Vert }_{L^{\infty }(0,T;V)}+\Vert u^{\prime \prime }{\Vert }_{L^2(0,T;V^{\ast })}\le C(\Vert f{\Vert }_{L^2(0,T;H)}+\Vert g{\Vert }_V+\Vert h{\Vert }_H),$$ with $C=C(\theta ,\Lambda ,\gamma ,T)$ and the time-coefficient bound $\Vert {\partial }_t{a}^{ij}{\Vert }_{L^{\infty }}$ ^{[Evans 2010 §7.2] [Lions-Magenes 1972 Ch. 3-4]}.

Proof. Step 1 (the finite-dimensional system is solvable). Fix $m$. With $u_m(t)={\sum }_{k\le m}{d}_m^k(t)w_k$ and $(w_k,w_{\ell }{)}_H={\delta }_{k\ell }$, the Galerkin equations read ${\ddot{d}}_m^k(t)+{\sum }_{\ell \le m}B[w_{\ell },w_k;t]d_m^{\ell }(t)=(f(t),w_k{)}_H$, a linear second-order system ${\ddot{d}}_m=-E(t)d_m+F(t)$ with $E(t{)}_{k\ell }=B[w_{\ell },w_k;t]\in L^{\infty }(0,T)$ and $F(t{)}_k=(f(t),w_k{)}_H\in L^2(0,T)$. Rewriting as a first-order system in $(d_m,{\dot{d}}_m)$ and applying the Carathéodory existence theorem gives a unique $d_m\in H^2(0,T;{\mathbb{R}}^m)$ with the prescribed $d_m(0),{\dot{d}}_m(0)$, hence a unique $u_m$ with $u_m,u_m^{\prime },u_m^{\prime \prime }\in L^2(0,T;V_m)$.

Step 2 (energy estimate, uniform in $m$). Multiply the $k$-th Galerkin equation by $(d_m^k{)}^{\prime }(t)$ and sum over $k\le m$; since $u_m^{\prime }(t)\in V_m$ this is the test choice $v=u_m^{\prime }(t)$: $$(u_m^{\prime \prime }(t),u_m^{\prime }(t){)}_H+B[u_m(t),u_m^{\prime }(t);t]=(f(t),u_m^{\prime }(t){)}_H.$$ The first term is $\frac{d}{dt}\frac{1}{2}\Vert u_m^{\prime }(t){\Vert }_H^2$. For the second, symmetry of $B$ gives $$\frac{d}{dt}\frac{1}{2}B[u_m(t),u_m(t);t]=B[u_m^{\prime }(t),u_m(t);t]+\frac{1}{2}{B}^{\prime }[u_m(t),u_m(t);t],$$ where $B^{\prime }[\cdot ,\cdot ;t]$ uses ${\partial }_t{a}^{ij}$ and ${\partial }_{t}c$; hence $B[u_m,u_m^{\prime };t]=\frac{d}{dt}\frac{1}{2}B[u_m,u_m;t]-\frac{1}{2}{B}^{\prime }[u_m,u_m;t]$. Writing the kinetic-plus-form energy $E_m(t)=\frac{1}{2}\Vert u_m^{\prime }(t){\Vert }_H^2+\frac{1}{2}B[u_m(t),u_m(t);t]$, $$E_m^{\prime }(t)=(f(t),u_m^{\prime }(t){)}_H+\frac{1}{2}{B}^{\prime }[u_m(t),u_m(t);t].$$ Estimate the right side: $(f,u_m^{\prime }{)}_H\le \frac{1}{2}\Vert f{\Vert }_H^2+\frac{1}{2}\Vert u_m^{\prime }{\Vert }_H^2$ by Young, and $|B^{\prime }[u_m,u_m;t]|\le \Vert {\partial }_t{a}^{ij}{\Vert }_{L^{\infty }}\Vert u_m{\Vert }_V^2+\Vert {\partial }_{t}c{\Vert }_{L^{\infty }}\Vert u_m{\Vert }_H^2\le C_1\Vert u_m{\Vert }_V^2$. By Gårding, $\Vert u_m{\Vert }_V^2\le \frac{1}{\beta }(B[u_m,u_m;t]+\gamma \Vert u_m{\Vert }_H^2)\le \frac{1}{\beta }(2E_m+\gamma \Vert u_m{\Vert }_H^2)$, and $\Vert u_m^{\prime }{\Vert }_H^2\le 2E_m$. Collecting, and using $\Vert u_m(t){\Vert }_H^2\le (\Vert u_m(0){\Vert }_H+{\int }_0^t\Vert u_m^{\prime }{\Vert }_H{)}^2\le C_2(1+{\int }_0^t{E}_m)$ from the fundamental theorem, $$E_m^{\prime }(t)\le \frac{1}{2}\Vert f(t){\Vert }_H^2+C_3{E}_m(t)+C_3{\int }_0^t{E}_m(s)ds+C_3.$$ Setting ${\Phi }_m(t)=E_m(t)+{\int }_0^t{E}_m$ and absorbing, ${\Phi }_m^{\prime }(t)\le \frac{1}{2}\Vert f(t){\Vert }_H^2+C_4{\Phi }_m(t)+C_4$, so by the integral Grönwall inequality ^{[Grönwall 1919]}, $$E_m(t)\le {\Phi }_m(t)\le e^{C_{4}T}(E_m(0)+\frac{1}{2}\Vert f{\Vert }_{L^2(0,T;H)}^2+C_{4}T),0\le t\le T.$$ Finally $E_m(0)=\frac{1}{2}\Vert u_m^{\prime }(0){\Vert }_H^2+\frac{1}{2}B[u_m(0),u_m(0);0]\le \frac{1}{2}\Vert h{\Vert }_H^2+\frac{\Lambda }{2}\Vert g{\Vert }_V^2$, since $u_m(0),u_m^{\prime }(0)$ are $H$-projections of $g,h$ onto $V_m$. From $E_m$ and coercivity, $\Vert u_m^{\prime }{\Vert }_{L^{\infty }(0,T;H)}$ and $\Vert u_m{\Vert }_{L^{\infty }(0,T;V)}$ are bounded uniformly in $m$ by $C(\Vert f{\Vert }_{L^2(0,T;H)}+\Vert g{\Vert }_V+\Vert h{\Vert }_H)$.

Step 3 (estimate on the second time-derivative). Fix $v\in V$, $\Vert v{\Vert }_V\le 1$, split $v=v_1+v_2$ with $v_1\in V_m$ the $H$-orthogonal projection, $\Vert v_1{\Vert }_V\le 1$. Then $(u_m^{\prime \prime }(t),v{)}_H=(u_m^{\prime \prime }(t),v_1{)}_H=(f(t),v_1{)}_H-B[u_m(t),v_1;t]$, so $|(u_m^{\prime \prime }(t),v{)}_H|\le \Vert f(t){\Vert }_H+\Lambda \Vert u_m(t){\Vert }_V$. Taking the supremum identifies $u_m^{\prime \prime }(t)\in V^{\ast }$ with $\Vert u_m^{\prime \prime }(t){\Vert }_{V^{\ast }}\le \Vert f(t){\Vert }_H+\Lambda \Vert u_m(t){\Vert }_V$, whence $\Vert u_m^{\prime \prime }{\Vert }_{L^2(0,T;V^{\ast })}\le \Vert f{\Vert }_{L^2(0,T;H)}+\Lambda \sqrt{T}\Vert u_m{\Vert }_{L^{\infty }(0,T;V)}$, bounded uniformly in $m$.

Step 4 (passage to the limit). By Steps 2-3 the sequences $u_m$, $u_m^{\prime }$, $u_m^{\prime \prime }$ are bounded in $L^{\infty }(0,T;V)$, $L^{\infty }(0,T;H)$, $L^2(0,T;V^{\ast })$. The first two are duals of separable spaces, so by Banach-Alaoglu (weak-* sequential compactness) and reflexivity of $L^2(0,T;V^{\ast })$ a subsequence satisfies $u_m\stackrel{\ast }{\rightharpoonup }u$ in $L^{\infty }(0,T;V)$, $u_m^{\prime }\stackrel{\ast }{\rightharpoonup }{u}^{\prime }$ in $L^{\infty }(0,T;H)$, and $u_m^{\prime \prime }\rightharpoonup u^{\prime \prime }$ in $L^2(0,T;V^{\ast })$ (weak limits of derivatives are derivatives of the weak limit). Fix $N$, $\psi \in C^2([0,T])$ with $\psi (T)={\psi }^{\prime }(T)=0$, and $w_k$ with $k\le N$. For $m\ge N$, multiply the Galerkin identity by $\psi$ and integrate by parts twice in time: $${\int }_0^T[(u_m,w_k{)}_H{\psi }^{\prime \prime }+B[u_m,w_k;t]\psi ]dt={\int }_0^T(f,w_k{)}_H\psi dt-(u_m^{\prime }(0),w_k{)}_H\psi (0)+(u_m(0),w_k{)}_H{\psi }^{\prime }(0).$$ Every term is linear and continuous for the weak-* topologies, so letting $m\to \infty$ replaces $u_m$ by $u$, $u_m(0)$ by $g$, $u_m^{\prime }(0)$ by $h$ (projections converge in $H$). The resulting identity holds for all $w_k$, by density for all $v\in V$, and undoing the integration by parts gives $⟨u^{\prime \prime }(t),v⟩+B[u(t),v;t]=(f(t),v{)}_H$ a.e. with $u(0)=g$, $u^{\prime }(0)=h$. The a priori estimate is the limit of the uniform bounds by weak-* lower semicontinuity.

Step 5 (uniqueness). Let $w$ be the difference of two solutions, so $w^{\prime \prime }+L(t)w=0$, $w(0)=0$, $w^{\prime }(0)=0$. The pairing identity (Proposition 4 below) makes $t\mapsto E(t)=\frac{1}{2}\Vert w^{\prime }(t){\Vert }_H^2+\frac{1}{2}B[w(t),w(t);t]$ absolutely continuous with $E^{\prime }(t)=⟨w^{\prime \prime }+Lw,w^{\prime }⟩+\frac{1}{2}{B}^{\prime }[w,w;t]=\frac{1}{2}{B}^{\prime }[w,w;t]$, since the first pairing vanishes by the equation. Hence $|E^{\prime }(t)|\le C\Vert w(t){\Vert }_V^2\le C^{\prime }(E(t)+\Vert w(t){\Vert }_H^2)$, and with $\Vert w(t){\Vert }_H^2\le ({\int }_0^t\Vert w^{\prime }{\Vert }_H{)}^2\le t{\int }_0^{t}2E$, Grönwall on $E(t)+{\int }_0^{t}E$ starting from $E(0)=0$ forces $E\equiv 0$, so $w^{\prime }\equiv 0$ and, with $w(0)=0$, $w\equiv 0$. $\square$

Bridge. The hyperbolic energy estimate is the foundational reason the construction converges, and it is the Gårding boundedness and coercivity of 02.16.04 read through the conserved quantity $E_m=\frac{1}{2}\Vert u_m^{\prime }{\Vert }_H^2+\frac{1}{2}B[u_m,u_m]$ rather than the dissipative one of 02.18.01: testing with the velocity $u_m^{\prime }$ replaces testing with $u_m$, so the time-integral of the $V$-norm is now controlled through the form energy and the $H$-norm of the velocity is propagated by the data. This is exactly the parabolic scheme of 02.18.01 with dissipation traded for conservation: where the parabolic energy fell monotonically, the hyperbolic energy is merely Grönwall-bounded, and the loss of monotonicity is the precise statement that there is no smoothing. The construction builds toward the semigroup picture of 02.18.03, where the first-order reduction $(u,u_t)$ generates a unitary-up-to-Grönwall group rather than a contraction semigroup, and the finite-propagation-speed theorem below generalises the constant-coefficient backward-light-cone uniqueness of 02.13.04 to variable coefficients. Putting these together, the central insight is that a hyperbolic equation is an elliptic energy identity tested against the velocity and closed by Grönwall, so existence again costs no explicit kernel; this appears again in the localized form, where restricting the same energy to a shrinking cone yields the domain-of-dependence statement that the wave equation respects causality.

Exercises Intermediate+

Advanced results Master

The Galerkin/energy existence theorem for the hyperbolic problem sits inside a wider structure that runs in close parallel to the parabolic theory of 02.18.01, with conservation everywhere replacing dissipation. The first-order reduction recasts the autonomous case as a group rather than a semigroup; the local-energy argument upgrades the global estimate to the finite-propagation-speed and domain-of-dependence theorems; the symmetric-hyperbolic formalism of Friedrichs embeds the second-order equation into a first-order system to which the same energy method applies verbatim; improved regularity is bought by differencing the equation in time rather than by parabolic smoothing; and the eigenfunction expansion makes the autonomous solution an exact superposition of oscillating modes.

Theorem 1 (first-order reduction and the propagator group; autonomous case). Let $L\ge 0$ be time-independent, self-adjoint with form domain $V$. Writing $U=(u,u_t)$ on the energy space $\mathcal{H}=V\times H$, the equation becomes $U^{\prime }=\mathcal{A}U+(0,f)$ with \mathcal{A}=\begin{psmallmatrix}0&I\\-L&0\end{psmallmatrix}, and $\mathcal{A}$ generates, by the Stone/Hille-Yosida theory of 02.18.03, a strongly continuous group $\{e^{t\mathcal{A}}{\}}_{t\in \mathbb{R}}$ that is unitary on $\mathcal{H}$ in the energy norm $\Vert (u,v){\Vert }_{\mathcal{H}}^2=B[u,u]+\Vert v{\Vert }_H^2$. The Galerkin solution coincides with the Duhamel/variation-of-parameters formula $U(t)=e^{t\mathcal{A}}(g,h)+{\int }_0^t{e}^{(t-s)\mathcal{A}}(0,f(s))ds$, the constant-coefficient instance being the d'Alembert/Kirchhoff representation of 02.13.04. Unitarity is the operator form of exact energy conservation; reversibility in $t$ (a group, not a one-sided semigroup) is the operator form of time-reversibility, the contrast with the analytic contraction semigroup $e^{-tL}$ of the parabolic case.

Theorem 2 (finite propagation speed and domain of dependence). Let $u$ be the weak solution with $L(t)$ symmetric uniformly elliptic, $\sum a^{ij}{\xi }_i{\xi }_j\le {\Lambda }_0|\xi {|}^2$, and set $\kappa =\sqrt{{\Lambda }_0}$. If $g$ and $h$ vanish on the ball $B_R(x_0)$ and $f$ vanishes on the spacetime cone $\{(x,t):|x-x_0|<R-\kappa t\}$, then $u\equiv 0$ on that cone; equivalently, $u(x_0,t_0)$ depends only on the data in $B_{\kappa t_0}(x_0)$ ^{[Courant-Friedrichs-Lewy 1928] [Evans 2010 §7.2]}. The proof is the local-energy flux inequality of Exercise 8 applied to the difference of two solutions: the local energy $e(t)=\frac{1}{2}{\int }_{|x-x_0|<R-\kappa t}(u_t^2+a^{ij}{\partial }_{i}u{\partial }_{j}u+u^2)dx$ is nonincreasing once the cone slope $\kappa$ exceeds the supremal characteristic speed, and vanishing initial local energy then propagates up the cone. The speed $\kappa$ is intrinsic to the coefficients, the variable-coefficient replacement for the single constant $c$ of 02.13.04; sharper bounds use the pointwise top eigenvalue of $(a^{ij}(x,t))$ to draw a curved cone of locally varying slope.

Theorem 3 (symmetric-hyperbolic embedding; Friedrichs 1954). Introducing $v=u_t$ and $p=\nabla u$, the scalar equation $u_{tt}-\sum {\partial }_i(a^{ij}{\partial }_{j}u)=f$ becomes a first-order system ${\partial }_{t}W+{\sum }_i{A}^i{\partial }_{i}W=g(W)$ for $W=(v,p)$ with the $A^i$ symmetric matrices ^{[Friedrichs 1954]}. Friedrichs' theory of symmetric hyperbolic systems gives existence, uniqueness, and the energy estimate directly from the symmetry of the $A^i$ by the same integration-by-parts that produced the scalar energy here, and the finite propagation speed is read off as the maximal eigenvalue of ${\sum }_i{A}^i{\nu }_i$ over unit covectors $\nu$. This is the framework in which Maxwell's equations, linearized elasticity, and the linearized Einstein equations are all hyperbolic, and it is the natural home for the cone-of-dependence geometry of Theorem 2.

Theorem 4 (improved regularity). If in addition $g\in V\cap H^2(\Omega )$ with $L(0)g\in H$, $h\in V$, $f\in H^1(0,T;H)$, and the coefficients are $C^1$ in time, then the weak solution satisfies $u\in L^{\infty }(0,T;H^2\cap V)$, $u^{\prime }\in L^{\infty }(0,T;V)$, $u^{\prime \prime }\in L^{\infty }(0,T;H)$, with the second-energy estimate $${\mathrm{ess}\sup }_{[0,T]}(\Vert u^{\prime \prime }{\Vert }_H^2+\Vert u^{\prime }{\Vert }_V^2)\le C(\Vert f{\Vert }_{H^1(0,T;H)}^2+\Vert L(0)g{\Vert }_H^2+\Vert h{\Vert }_V^2)$$ ^{[Evans 2010 §7.2] [Ladyzhenskaya 1985 Ch. V]}. The mechanism is to differentiate the Galerkin equation in time and apply the first energy estimate to $\tilde{u}=u^{\prime }$, testing with ${\tilde{u}}^{\prime }=u^{\prime \prime }$; unlike the parabolic gain of 02.18.01, where testing with $u^{\prime }$ traded one time-derivative for one elliptic derivative through smoothing, here each differentiation in time costs one degree of regularity in the data, and there is no free gain. Higher regularity follows by repeated time-differencing plus elliptic regularity of 02.16.04, recovering the classical solution under compatibility conditions.

Theorem 5 (eigenfunction expansion; autonomous case). For $L=-\sum {\partial }_i(a^{ij}{\partial }_j\cdot )$ time-independent with Dirichlet eigenpairs $({\mu }_k,w_k)$, ${\mu }_k>0$, the solution of $u_{tt}+Lu=0$, $u(0)=g$, $u_t(0)=h$, is the exact superposition $$u(x,t)=\sum _{k\ge 1}[(g,w_k{)}_H\cos (\sqrt{{\mu }_k}t)+\frac{(h,w_k{)}_H}{\sqrt{{\mu }_k}}\sin (\sqrt{{\mu }_k}t)]w_k(x),$$ with energy $E=\frac{1}{2}{\sum }_k({\mu }_k|(g,w_k){|}^2+|(h,w_k){|}^2)$ conserved term by term, each mode an undamped oscillator of frequency $\sqrt{{\mu }_k}$. This is the bounded-domain analogue of the Fourier representation behind d'Alembert's formula 02.13.04, and the convergence of the partial sums is the Galerkin convergence of the main theorem made explicit in the eigenbasis.

Synthesis. The hyperbolic energy estimate is the foundational reason the Galerkin scheme converges, and it is exactly the Gårding boundedness of 02.16.04 read through the conserved energy $E=\frac{1}{2}\Vert u^{\prime }{\Vert }_H^2+\frac{1}{2}B[u,u]$, tested against the velocity $u^{\prime }$ rather than the parabolic $u$: the kinetic norm of the velocity is propagated and the form energy controls the $V$-norm, so existence costs no kernel and the construction is dual to the elliptic theory in the same precise sense as the parabolic case, the frozen-time Lax-Milgram solvability making the Galerkin matrix invertible at each instant. This is exactly the parabolic scheme of 02.18.01 with one structural substitution, dissipation replaced by conservation: where the parabolic energy fell and bought smoothing, the hyperbolic energy is conserved and buys none, and the loss of the monotone sign is the central insight that hyperbolic regularity is exactly the regularity of the data, propagated.

Putting these together, the same energy identity generalises in two directions at once. Localized to a shrinking cone it gives the finite-propagation-speed theorem, the variable-coefficient promotion of the constant-speed backward-light-cone of 02.13.04, so the bridge from this unit reaches back to the d'Alembert causality it generalises; reduced to first order it gives the unitary propagator group of 02.18.03, the conservative twin of the analytic semigroup, so the bridge reaches forward to the generation theory the conserved energy feeds. The central insight is that one symmetric energy identity, tested against the velocity and closed by Grönwall, simultaneously yields existence, uniqueness, the absence of smoothing, and the cone of dependence, the four facts that distinguish hyperbolic evolution from its parabolic sibling.

Full proof set Master

Proposition 1 (uniform hyperbolic energy estimate). Under the hypotheses of the main theorem the Galerkin approximations satisfy ${\sup }_m(\Vert u_m^{\prime }{\Vert }_{L^{\infty }(0,T;H)}+\Vert u_m{\Vert }_{L^{\infty }(0,T;V)}+\Vert u_m^{\prime \prime }{\Vert }_{L^2(0,T;V^{\ast })})\le C(\Vert f{\Vert }_{L^2(0,T;H)}+\Vert g{\Vert }_V+\Vert h{\Vert }_H)$.

Proof. Testing the Galerkin identity with $v=u_m^{\prime }(t)$ gives $\frac{d}{dt}\frac{1}{2}\Vert u_m^{\prime }{\Vert }_H^2+B[u_m,u_m^{\prime };t]=(f,u_m^{\prime }{)}_H$. Symmetry of $B$ yields $B[u_m,u_m^{\prime };t]=\frac{d}{dt}\frac{1}{2}B[u_m,u_m;t]-\frac{1}{2}{B}^{\prime }[u_m,u_m;t]$, so the energy $E_m=\frac{1}{2}\Vert u_m^{\prime }{\Vert }_H^2+\frac{1}{2}B[u_m,u_m;t]$ obeys $E_m^{\prime }=(f,u_m^{\prime }{)}_H+\frac{1}{2}{B}^{\prime }[u_m,u_m;t]$. Young bounds $(f,u_m^{\prime }{)}_H\le \frac{1}{2}\Vert f{\Vert }_H^2+\frac{1}{2}\Vert u_m^{\prime }{\Vert }_H^2\le \frac{1}{2}\Vert f{\Vert }_H^2+E_m$, and $|B^{\prime }[u_m,u_m;t]|\le C_1\Vert u_m{\Vert }_V^2\le \frac{C_1}{\beta }(2E_m+\gamma \Vert u_m{\Vert }_H^2)$ by Gårding. With $\Vert u_m(t){\Vert }_H^2\le 2(\Vert u_m(0){\Vert }_H^2+t{\int }_0^t\Vert u_m^{\prime }{\Vert }_H^2)\le C_2(1+{\int }_0^t{E}_m)$, set ${\Phi }_m=E_m+{\int }_0^t{E}_m$ to get ${\Phi }_m^{\prime }\le \frac{1}{2}\Vert f{\Vert }_H^2+C_3{\Phi }_m+C_3$. Grönwall (Exercise 4) gives $E_m(t)\le {\Phi }_m(t)\le e^{C_{3}T}(E_m(0)+\frac{1}{2}\Vert f{\Vert }_{L^2(0,T;H)}^2+C_{3}T)$, and $E_m(0)\le \frac{1}{2}\Vert h{\Vert }_H^2+\frac{\Lambda }{2}\Vert g{\Vert }_V^2$. Coercivity converts the $E_m$-bound into bounds on $\Vert u_m^{\prime }{\Vert }_{L^{\infty }(0,T;H)}$ and $\Vert u_m{\Vert }_{L^{\infty }(0,T;V)}$. Finally $\Vert u_m^{\prime \prime }(t){\Vert }_{V^{\ast }}\le \Vert f(t){\Vert }_H+\Lambda \Vert u_m(t){\Vert }_V$ (Step 3 of the theorem), so $\Vert u_m^{\prime \prime }{\Vert }_{L^2(0,T;V^{\ast })}\le \Vert f{\Vert }_{L^2(0,T;H)}+\Lambda \sqrt{T}\Vert u_m{\Vert }_{L^{\infty }(0,T;V)}$. $\square$

Proposition 2 (existence of a weak solution). A subsequence of $(u_m)$ converges weak-* in $L^{\infty }(0,T;V)$ with $u_m^{\prime }\stackrel{\ast }{\rightharpoonup }{u}^{\prime }$ in $L^{\infty }(0,T;H)$ and $u_m^{\prime \prime }\rightharpoonup u^{\prime \prime }$ in $L^2(0,T;V^{\ast })$ to a weak solution $u$ with $u(0)=g$, $u^{\prime }(0)=h$.

Proof. Proposition 1 and the separability of $V,H$ give, via Banach-Alaoglu, a subsequence with $u_m\stackrel{\ast }{\rightharpoonup }u$ in $L^{\infty }(0,T;V)$, $u_m^{\prime }\stackrel{\ast }{\rightharpoonup }\chi$ in $L^{\infty }(0,T;H)$, $u_m^{\prime \prime }\rightharpoonup \xi$ in $L^2(0,T;V^{\ast })$. For $\phi \in C_c^{\infty }(0,T)$, $v\in V$: $\int {\phi }^{\prime }(u_m,v{)}_H=-\int \phi (u_m^{\prime },v{)}_H$ and $\int {\phi }^{\prime }(u_m^{\prime },v{)}_H=-\int \phi ⟨u_m^{\prime \prime },v⟩$; passing to the limit identifies $\chi =u^{\prime }$ and $\xi =u^{\prime \prime }$, so $u\in \mathcal{W}(0,T)$. Fix $k$, $\psi \in C^2([0,T])$ with $\psi (T)={\psi }^{\prime }(T)=0$. For $m\ge k$, integrating the Galerkin identity twice by parts in time gives ${\int }_0^T[(u_m,w_k{)}_H{\psi }^{\prime \prime }+B[u_m,w_k;t]\psi ]dt={\int }_0^T(f,w_k{)}_H\psi dt-(u_m^{\prime }(0),w_k{)}_H\psi (0)+(u_m(0),w_k{)}_H{\psi }^{\prime }(0)$. Each term is weak-* continuous in $u_m$, and $u_m(0)\to g$, $u_m^{\prime }(0)\to h$ in $H$, so the limit reads with $u,g,h$ in place of $u_m,u_m(0),u_m^{\prime }(0)$. Density of $\{w_k\}$ extends this to all $v\in V$; choosing $\psi \in C_c^{\infty }(0,T)$ recovers the equation a.e., and comparing the boundary terms for general $\psi$ forces $u(0)=g$, $u^{\prime }(0)=h$. $\square$

Proposition 3 (uniqueness and continuous dependence). The weak solution is unique, and $\Vert u^{\prime }{\Vert }_{C([0,T];H)}+\Vert u{\Vert }_{C([0,T];V)}\le C(\Vert f{\Vert }_{L^2(0,T;H)}+\Vert g{\Vert }_V+\Vert h{\Vert }_H)$.

Proof. The estimate is the weak-* lower-semicontinuous limit of Proposition 1, with the continuity in time supplied by the regularity $u\in C([0,T];V)$, $u^{\prime }\in C([0,T];H)$ (Proposition 4). For uniqueness, the difference $w$ of two solutions with equal data solves $w^{\prime \prime }+Lw=0$, $w(0)=0$, $w^{\prime }(0)=0$. By Proposition 4 the energy $E(t)=\frac{1}{2}\Vert w^{\prime }{\Vert }_H^2+\frac{1}{2}B[w,w;t]$ is absolutely continuous with $E^{\prime }(t)=⟨w^{\prime \prime }+Lw,w^{\prime }⟩+\frac{1}{2}{B}^{\prime }[w,w;t]=\frac{1}{2}{B}^{\prime }[w,w;t]$, the pairing $⟨w^{\prime \prime }+Lw,w^{\prime }⟩$ vanishing by the equation. Then $|E^{\prime }|\le C\Vert w{\Vert }_V^2\le C^{\prime }(E+\Vert w{\Vert }_H^2)$ and $\Vert w(t){\Vert }_H^2\le t{\int }_0^t\Vert w^{\prime }{\Vert }_H^2\le 2t{\int }_0^{t}E$, so $\frac{d}{dt}(E+{\int }_0^{t}E)\le C^{\prime \prime }(E+{\int }_0^{t}E)$; Grönwall from $E(0)=0$ gives $E\equiv 0$, hence $w^{\prime }\equiv 0$ and $w\equiv 0$. $\square$

Proposition 4 (regularity in time and the energy identity). Every weak solution $u$ with $u\in L^{\infty }(0,T;V)$, $u^{\prime }\in L^{\infty }(0,T;H)$, $u^{\prime \prime }\in L^2(0,T;V^{\ast })$ has representatives $u\in C([0,T];V)$, $u^{\prime }\in C([0,T];H)$, and $t\mapsto E(t)=\frac{1}{2}\Vert u^{\prime }(t){\Vert }_H^2+\frac{1}{2}B[u(t),u(t);t]$ is absolutely continuous with $E^{\prime }(t)=⟨u^{\prime \prime }(t)+L(t)u(t),u^{\prime }(t)⟩+\frac{1}{2}{B}^{\prime }[u(t),u(t);t]$ for a.e. $t$.

Proof. The strong continuity $u\in C([0,T];V)$, $u^{\prime }\in C([0,T];H)$ is not automatic from the function-space memberships and is proved by a mollification-in-time argument (Lions-Magenes): regularize $u$ to $u_{ϵ}$ smooth in $t$, for which the classical product rule gives $\frac{d}{dt}{E}_{ϵ}=(u_{ϵ}^{\prime \prime },u_{ϵ}^{\prime }{)}_H+B[u_{ϵ},u_{ϵ}^{\prime };t]+\frac{1}{2}{B}^{\prime }[u_{ϵ},u_{ϵ};t]=(u_{ϵ}^{\prime \prime }+L{u}_{ϵ},u_{ϵ}^{\prime }{)}_{?}+\frac{1}{2}{B}^{\prime }[u_{ϵ},u_{ϵ};t]$. The right-hand side converges in $L^1(0,T)$ as $ϵ\to 0$ because $u_{ϵ}^{\prime \prime }+L{u}_{ϵ}\to u^{\prime \prime }+Lu$ in $L^2(0,T;V^{\ast })$ and $u_{ϵ}^{\prime }\to u^{\prime }$ in $L^2(0,T;V)\cap C([0,T];H)$ (the latter is where the regularity claim is used and proved, via the boundedness of the mollified energies and an Ascoli argument in $C([0,T];H_{\text{weak}})$ upgraded to strong continuity by the continuity of $t\mapsto E(t)$). Integrating $\frac{d}{dt}{E}_{ϵ}$ and passing to the limit yields the absolute continuity of $E$ and the stated derivative; the same passage gives the pointwise-in-time values defining $u(0)\in V$, $u^{\prime }(0)\in H$. $\square$

Proposition 5 (finite propagation speed, variable coefficients). Let $u$ solve $u^{\prime \prime }+L(t)u=0$ weakly with $L(t)$ symmetric uniformly elliptic, $\sum a^{ij}{\xi }_i{\xi }_j\le {\Lambda }_0|\xi {|}^2$, and let $\kappa =\sqrt{{\Lambda }_0}$. If $g$ and $h$ vanish on $B_R(x_0)$, then $u\equiv 0$ on the cone $\mathcal{C}=\{(x,t):0\le t<R/\kappa ,|x-x_0|<R-\kappa t\}$.

Proof. By Proposition 4 (applied locally, using that $u$ is, for a.e. $t$, an $H^1$ function and that the local energy below is finite and absolutely continuous in $t$) define $e(t)=\frac{1}{2}{\int }_{|x-x_0|<R-\kappa t}(u_t^2+a^{ij}{\partial }_{i}u{\partial }_{j}u)dx$. Differentiating with the domain shrinking at radial speed $\kappa$, $$e^{\prime }(t)={\int }_{B(t)}(u_t{u}_{tt}+a^{ij}{\partial }_{i}u{\partial }_j{u}_t+\frac{1}{2}({\partial }_t{a}^{ij}){\partial }_{i}u{\partial }_{j}u)dx-\frac{\kappa }{2}{\int }_{\partial B(t)}(u_t^2+a^{ij}{\partial }_{i}u{\partial }_{j}u)dS.$$ Using $u_{tt}={\partial }_i(a^{ij}{\partial }_{j}u)$ a.e. and integrating the first interior term by parts, ${\int }_{B(t)}{u}_t{\partial }_i(a^{ij}{\partial }_{j}u)dx=-{\int }_{B(t)}{a}^{ij}{\partial }_{j}u{\partial }_i{u}_{t}dx+{\int }_{\partial B(t)}{u}_t{a}^{ij}{\partial }_{j}u{\nu }_{i}dS$; the interior gradient terms cancel against $\int a^{ij}{\partial }_{i}u{\partial }_j{u}_t$ (symmetry $a^{ij}=a^{ji}$). Thus $$e^{\prime }(t)={\int }_{\partial B(t)}{u}_t{a}^{ij}{\partial }_{j}u{\nu }_{i}dS-\frac{\kappa }{2}{\int }_{\partial B(t)}(u_t^2+a^{ij}{\partial }_{i}u{\partial }_{j}u)dS+\frac{1}{2}{\int }_{B(t)}({\partial }_t{a}^{ij}){\partial }_{i}u{\partial }_{j}udx.$$ On $\partial B(t)$, $|u_t{a}^{ij}{\partial }_{j}u{\nu }_i|\le \sqrt{{\Lambda }_0}|u_t|\sqrt{a^{ij}{\partial }_{i}u{\partial }_{j}u}\le \frac{\sqrt{{\Lambda }_0}}{2}(u_t^2+a^{ij}{\partial }_{i}u{\partial }_{j}u)$ by the $a$-Cauchy-Schwarz bound $a(\nu ,\nu )\le {\Lambda }_0$ and Young. With $\kappa =\sqrt{{\Lambda }_0}$ the boundary integrals cancel up to sign, leaving $e^{\prime }(t)\le \frac{1}{2}{\int }_{B(t)}({\partial }_t{a}^{ij}){\partial }_{i}u{\partial }_{j}udx\le \frac{\Vert {\partial }_t{a}^{ij}{\Vert }_{L^{\infty }}}{\theta }e(t)=:\omega e(t)$. Grönwall with $e(0)=0$ (data vanish on $B_R(x_0)$) gives $e(t)\le e^{\omega t}e(0)=0$, so $e\equiv 0$ on $[0,R/\kappa )$. Hence $u_t\equiv 0$ and $\nabla u\equiv 0$ on $\mathcal{C}$, and with the vanishing initial data $u\equiv 0$ on $\mathcal{C}$. $\square$

Connections Master

• The spatial engine is the elliptic weak theory of 02.16.04: the boundedness and Gårding coercivity proved there from uniform ellipticity are the exact inputs to the hyperbolic energy estimate, and the Dirichlet eigenbasis driving the Galerkin scheme and the eigenfunction expansion of Theorem 5 is the spectral decomposition of the compact resolvent built there. This unit owns the conservative time-evolution layer; 02.16.04 owns the frozen-time elliptic solvability the Galerkin matrix inherits at each instant.

• The parabolic sibling 02.18.01 supplies the entire Bochner-space and Galerkin apparatus reused here; the two units differ by exactly one structural substitution, dissipation against conservation. The parabolic energy is tested against $u$ and decays, buying smoothing; the hyperbolic energy is tested against $u_t$ and is conserved, buying none. Reading the two units side by side is the cleanest way to see why parabolic solutions are instantly smooth while hyperbolic solutions merely propagate their initial regularity.

• The constant-coefficient whole-space wave equation of 02.13.04 is the explicitly solvable special case: its d'Alembert, Kirchhoff, and Poisson formulas are the closed-form realizations of the abstract propagator group $e^{t\mathcal{A}}$ of Theorem 1, and its backward-light-cone uniqueness is the constant-speed instance of the finite-propagation-speed theorem proved here for variable coefficients. This unit is the variable-coefficient existence theory of which 02.13.04 is the one computable instance.

• The first-order reduction of Theorem 1 is developed in its own right in 02.18.03 (Hille-Yosida and $C_0$-semigroups), where the generator \mathcal{A}=\begin{psmallmatrix}0&I\\-L&0\end{psmallmatrix} is shown to generate a unitary group by Stone's theorem; the conserved hyperbolic energy of this unit is the unitarity hypothesis of that generation theorem, so the two units are the constructive and the generator-theoretic faces of the same conservative evolution.

• The variational/minimization theory of 02.18.04 (the direct method) finds elliptic critical points as energy stationary points; the hyperbolic flow of this unit is the formal Hamiltonian flow of that same Dirichlet energy, with $\frac{1}{2}\Vert u_t{\Vert }_H^2$ the kinetic term, so the conserved $E$ of this unit is the Hamiltonian whose stationary configurations are the variational solutions of 02.18.04. Time-evolution existence and variational existence are thereby the dynamic and the static faces of one energy functional within the chapter.

Historical & philosophical context Master

The extension of the Galerkin spatial-projection idea from steady-state to time-dependent problems, due to Sandro Faedo in his 1949 Annali della Scuola Normale Superiore di Pisa memoir ^{[Faedo 1949]}, applies verbatim to the hyperbolic case: project onto finitely many spatial modes, solve the resulting system of ordinary differential equations, and pass to the limit using a-priori energy bounds. For the second-order-in-time equation the bounds come from the conserved energy rather than the dissipated one, the only structural change from the parabolic construction. The functional-analytic completion in the Gelfand-triple framework belongs to Jacques-Louis Lions and Enrico Magenes, whose 1972 Non-Homogeneous Boundary Value Problems and Applications I ^{[Lions-Magenes 1972]} treats hyperbolic and parabolic evolution equations within one apparatus, proving the regularity-in-time and trace theorems that give the two hyperbolic initial conditions their meaning.

The finite-propagation-speed phenomenon was isolated in the 1928 Mathematische Annalen paper of Richard Courant, Kurt Friedrichs, and Hans Lewy ^{[Courant-Friedrichs-Lewy 1928]}, whose domain-of-dependence analysis, introduced to prove convergence of finite-difference schemes, identified the backward characteristic cone as the carrier of all information reaching a spacetime point. The same cone became the geometric object underlying Friedrichs' 1954 theory of symmetric hyperbolic systems ^{[Friedrichs 1954]}, which embeds the second-order scalar equation into a first-order system whose energy method and propagation speed are read directly from the symmetry and eigenvalues of the coefficient matrices, the framework that covers Maxwell, elasticity, and the linearized Einstein equations. The systematic energy-method treatment of mixed initial-boundary-value problems for hyperbolic equations on bounded domains is developed in Olga Ladyzhenskaya's Boundary Value Problems of Mathematical Physics ^{[Ladyzhenskaya 1985]}, and the modern lecture-note synthesis is Peter Lax's Hyperbolic Partial Differential Equations ^{[Lax 2006]}.

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