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Finite speed of propagation does not imply strong Huygens

Statement refuted

The finite-speed property (Compact support expands at speed at most c) coexists with three distinct wave phenomena: the strong Huygens principle can fail (The strong Huygens principle in the homogeneous Cauchy setting), regularity need not improve, and nonnegative displacement data need not produce a nonnegative solution. The following compactly supported witnesses make these distinctions explicit.

Failure of strong Huygens. Let $c > 0$ , $x_{0} \in R^{n}$ , $t_{0} > 0$ and $n \in {1, 2}$ . In dimension $1$ , choose $u_{0} = 0$ and a nonnegative nonzero $u_{1} \in C_{c}^{1} (R)$ supported in $(x_{0} - c t_{0}, x_{0} + c t_{0})$ . Then the d'Alembert formula gives $u (x_{0}, t_{0}) = \frac{1}{2 c} \int u_{1} > 0$ . In dimension $2$ , choose $u_{0} = 0$ and a nonnegative nonzero $u_{1} \in C_{c}^{2} (R^{2})$ supported in $B_{r} (x_{0})$ for some $r < c t_{0}$ ; the Poisson formula gives $u (x_{0}, t_{0}) > 0$ . In both cases the data vanish on a neighbourhood of $S (x_{0}, c t_{0})$ , yet the value is nonzero, while finite speed still holds.
No smoothing. There is a compactly supported $F \in C^{2} (R)$ that is not $C^{3}$ . The traveling wave $u (x, t) = F (x - c t)$ solves the one-dimensional equation and remains $C^{2}$ but not $C^{3}$ for every $t \geq 0$ . Its support translates at speed $c$ .
No maximum principle. In dimension $2$ , take a nonnegative nonzero $u_{0} \in C_{c}^{\infty} (B_{r} (x_{0}))$ and $u_{1} = 0$ . At any time $t_{0} > r / c$ , the displacement term of Poisson's formula gives $u (x_{0}, t_{0}) < 0$ . Thus the solution can change sign although the initial displacement is nonnegative and the initial velocity is zero.

All four data pairs are compactly supported. The d’Alembert witnesses extend $C^{2}$ through time zero; for the smooth Poisson witnesses the descended sphere expression $\partial_{t} [t M_{U_{0}}^{(3)} ((x, 0), c t)] + t M_{U_{1}}^{(3)} ((x, 0), c t)$ extends smoothly through zero, since the signed-radius means are integrals of smooth data over the fixed compact sphere and may be differentiated there on any compact parameter set. Thus the initial regularity hypothesis is met and Compact support expands at speed at most c supplies the finite-speed bound. The no-smoothing and sign-change examples are independent of the Huygens witnesses; they show why the qualitative properties listed by Hunter require separate arguments.

Facts & Assumptions

Given: ${A C}_{ω}$ ; $c > 0$ ; the compactly supported smooth data chosen in the proof; and a compactly supported $C^{2}$ profile.

[F1]

For $u_{0} \in C^{2} (R)$ and $u_{1} \in C^{1} (R)$ , the unique classical solution is $u (x, t) = \frac{1}{2} (u_{0} (x - c t) + u_{0} (x + c t)) + \frac{1}{2 c} \int_{x - c t}^{x + c t} u_{1} (y) d y .$ (d'Alembert's formula and uniqueness in one dimension)

[F2]

For $u_{0} \in C^{3} (R^{2})$ and $u_{1} \in C^{2} (R^{2})$ , the two-dimensional solution is the Poisson expression $u (x, t) = \frac{1}{2 π c} \frac{\partial}{\partial t} \int_{B_{c t} (x)} \frac{u_{0} (y)}{\sqrt{c^{2} t^{2} - ∣ y - x ∣^{2}}} d y + \frac{1}{2 π c} \int_{B_{c t} (x)} \frac{u_{1} (y)}{\sqrt{c^{2} t^{2} - ∣ y - x ∣^{2}}} d y .$ (Poisson's formula in two dimensions by descent)

[F3]

Finite propagation: for a $C^{2}$ solution defined on a neighbourhood of the initial slab, with data supported in a compact $K$ and a source supported in ${(x, t) : dist (x, K) \leq c t}$ (in particular for a zero source), $supp u (\cdot, t) \subseteq K + {\overline{B}}_{c t} (0)$ for every $t$ . (Compact support expands at speed at most c)

[F4]

The strong Huygens principle in the homogeneous Cauchy setting is the statement that the value is carried by the sphere $S (x_{0}, c t_{0}) = \partial B_{c t_{0}} (x_{0})$ : admissible perturbations vanishing on a neighbourhood of $S$ do not change $u (x_{0}, t_{0})$ . (The strong Huygens principle in the homogeneous Cauchy setting)

[F5]

For any centre and radius there is a smooth nonnegative bump equal to one on the concentric half-radius ball and supported inside the full ball, by translating A smooth bump between concentric Euclidean balls.

[F6]

A nonnegative integral is monotone and positively homogeneous; it vanishes exactly for functions zero almost everywhere. A continuous function positive somewhere is bounded below by a positive constant on a smaller ball of positive measure. (Monotonicity and nonnegative homogeneity of the nonnegative integral, A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere, Sphere and ball measures scale in Rn)

[F7]

A parameter derivative may pass under the integral when dominated by a fixed integrable function on the parameter interval. (Differentiation under the integral sign)

[F8]

Chain, product and real-power rules apply to the explicit profile off its join points; the integral of a continuous derivative is its endpoint difference, and Darboux, Riemann and Lebesgue interval integrals agree. (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Continuity and derivatives of positive-base real powers, The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, The Darboux and Riemann definitions agree: a bounded $f$ on $[a, b]$ is Darboux integrable with integral $I$ if and only if for every real $ε > 0$ there is a real $δ > 0$ such that $∣ S (f, P, ξ) - I ∣ < ε$ for every tagged partition of mesh below $δ$ , A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral)

Proof

1.1givenF1F3F4F5F6

The one-dimensional Huygens witness: choose $u_{0} = 0$ and a smooth nonnegative nonzero bump $u_{1}$ as in [F5] supported in $(x_{0} - c t_{0}, x_{0} + c t_{0})$ . Formula [F1] gives $u (x_{0}, t_{0}) = \frac{1}{2 c} \int_{x_{0} - c t_{0}}^{x_{0} + c t_{0}} u_{1} (y) d y > 0.$ The data are compactly supported inside the open base interval, hence vanish on a neighbourhood of its boundary sphere; [F3] supplies finite speed.

1.2givenF2F3F4F5F6

The two-dimensional Huygens witness: choose $0 < r < c t_{0}$ and the nonnegative nonzero smooth bump $u_{1}$ of [F5] supported in $B_{r} (x_{0})$ , with $u_{0} = 0$ . Formula [F2] gives $u (x_{0}, t_{0}) = \frac{1}{2 π c} \int_{B_{r} (x_{0})} \frac{u_{1} (y)}{\sqrt{c^{2} t_{0}^{2} - ∣ y - x_{0} ∣^{2}}} d y > 0.$ Again the data vanish on a neighbourhood of $S (x_{0}, c t_{0})$ , and [F3] applies.

1.3givenF1F8

Compact traveling wave with no smoothing: define $F (s) : = {\begin{cases} (1 - s^{2})^{3} ∣ s ∣^{5 / 2}, & ∣ s ∣ < 1, \\ 0, & ∣ s ∣ \geq 1. \end{cases}$ At $s = \pm 1$ the factor $(1 - s^{2})^{3}$ makes $F, F^{'}, F^{''}$ tend to zero, so the extension by zero is compactly supported and $C^{2}$ . The power and product rules [F8] give $F \to 0$ and $F^{'} (s) = \frac{5}{2} sgn (s) ∣ s ∣^{3 / 2} + O (∣ s ∣^{7 / 2}) \to 0$ as $s \to 0$ . The difference quotients of $F$ and $F^{'}$ at zero tend to zero, so $F^{'} (0) = F^{''} (0) = 0$ . Near $s = 0$ , $F^{''} (s) = \frac{15}{4} ∣ s ∣^{1 / 2} + O (∣ s ∣^{5 / 2}),$ so $F^{''} (h) / h \to + \infty$ as $h ↓ 0$ ; hence $F$ is not $C^{3}$ . Take $u_{0} = F$ , $u_{1} = - c F^{'}$ . In [F1], by the FTC in [F8], the integral term equals $- \frac{1}{2} (F (x + c t) - F (x - c t))$ , so the solution simplifies to $u (x, t) = F (x - c t)$ . It is $C^{2}$ and not $C^{3}$ at $x = c t$ , and its compact support is translated exactly at speed $c$ .

1.4givenF2F3F5F6F7

Nonnegative displacement becomes negative: take the nonnegative nonzero smooth bump $u_{0}$ of [F5] supported in $B_{r} (x_{0})$ , $u_{1} = 0$ , and $t_{0} > r / c$ . Since the support is strictly inside $B_{c t} (x_{0})$ for $t$ near $t_{0}$ , on a small closed time interval about $t_{0}$ the quantity $c^{2} t^{2} - r^{2}$ has a positive lower bound. Thus the integrand and its time derivative are uniformly bounded on the compact support, providing a constant integrable majorant; [F7] permits differentiating the displacement integral in [F2] over the fixed support: $u (x_{0}, t_{0}) = - \frac{c t_{0}}{2 π} \int_{B_{r} (x_{0})} \frac{u_{0} (y)}{(c^{2} t_{0}^{2} - ∣ y - x_{0} ∣^{2})^{3 / 2}} d y < 0.$ The strict inequality follows by [F6] because the kernel is positive and $u_{0}$ is nonnegative and nonzero. Thus positivity is not preserved, despite nonnegative displacement and zero initial velocity; [F3] still gives finite speed.

2.1step 1.1step 1.2F3F4

Huygens conclusion: in steps 1.1 and 1.2 each data pair is supported strictly inside the relevant base ball, so it agrees with the zero pair on a neighbourhood of the sphere but gives a nonzero value at the vertex. This contradicts the defining data-insensitivity in [F4]. Finite propagation [F3] remains true; therefore finite speed does not imply strong Huygens, as recorded in Finite propagation is not the Huygens principle.

3.1step 1.3step 1.4∎

Regularity and positivity conclusions: step 1.3 retains a second-derivative cusp under translation, so the wave flow has no smoothing; step 1.4 gives an explicit failure of positivity preservation, hence of a maximum principle.

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Sources

Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript, AMS Graduate Studies in Mathematics) (standard reference, not scraped)
Jared Speck, MIT 18.152 Introduction to Partial Differential Equations, Class Meeting #12: Kirchhoff's Formula and Minkowskian Geometry (Fall 2011) (standard reference, not scraped)