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I am reading the paper Large global solutions for energy supercritical nonlinear wave equations on $\mathbb{R}^{3+1}$ by Krieger and Schlag and am confused by one of their steps.

For context, $v(t,r)$ is radial and solves a septic nonlinear wave equation $$-\partial_{tt}v + \Delta v = F(v),$$ on $(t,x)\in[T, T_1]\times \mathbb{R}^3$ where $F(v) = G(v) + g(r,t)$, where $G(0)=0$ and $g(r,t)$ is supported on $\{r=|x|<t+C\}$. We know that at time $T$, $v$ is supported in the ball $\{r=|x|\le T+C\}$ with $C>0$. The authors then claim, on page 34, that by Huygens' principle, it follows $v(r,t)$ is supported in the ball $\{r=|x|<t+C\}$ for each fixed $t\ge T$

However, I believe this simply follows from the finite propagation speed property of the wave equation. Is this correct, or is the Huygens' principle genuinely needed here?

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  • $\begingroup$ I apologize, I have fixed the title. $\endgroup$
    – Dispersion
    Commented 12 hours ago

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As a remark, invoquing the finite propagation speed property for the wave equation is hand waving. How do one make this statement a rigorous mathematical concept ?

For a general hyperbolic PDE (or system of PDEs), you can make a definition, by looking at the velocity of planar waves. But then you have to transform this definition into a theorem and this requires pseudo-differential calculus.

For the wave equation, or more generally for so-called symmetric hyperbolic (systems of) PDEs, a much simpler proof occurs, which consists in integrating the energy equation $$\partial_t\frac12(v_t^2+|\nabla v|^2)-{\rm div}(v_t\nabla v)=-v_tF(v)$$ over the characteristic cone $t\in(0,T)$, $|x|<t+c$.

Mind that you need of course $F(0)=0$.

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Q: Huygens' principle or finite speed of propagation?

It's the same thing, Huygens principle is a statement of causality, which means finite speed of wave front propagation.

Without Huygens principle you might imagine that the right-hand-side of the wave equation is a source term that allows a nonzero $v(x,t)$ to appear at $|x|>t+C$, without any relationship to the wavefronts at earlier times. The notion that a nonzero $v$ at some later time is causally related to a wavefront at an earlier time is the content of Huygens principle.

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  • $\begingroup$ According to en.wikipedia.org/wiki/Huygens%E2%80%93Fresnel_principle : In 1900, Jacques Hadamard observed that Huygens' principle was broken when the number of spatial dimensions is even. Does this mean that for even number of spatial dimensions the speed of wave front propagation is infinite ? $\endgroup$
    – jjcale
    Commented 28 mins ago
  • $\begingroup$ Indeed, this will not work in two spatial dimensions (the question is for $\mathbb{R}^3$). $\endgroup$ Commented 21 mins ago

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