Wave tails in one and even spatial dimensions: strong Huygens fails

The one-dimensional tail: with $u_0=0$, [F3] gives $u(x_0,t_0)=\frac{1}{2c}{\int }_{x_0-c{t}_0}^{x_0+c{t}_0}{u}_1$ for every admissible $u_1\in C_c^1(\mathbb{R})$, and if $u_1$ is supported in the open interval $(x_0-c{t}_0,x_0+c{t}_0)$ and has nonzero integral then this value is nonzero, while the zero data give $u\equiv 0$; the difference $(0,u_1)$ is supported strictly inside the base ball $(x_0-c{t}_0,x_0+c{t}_0)$, so this is a genuine failure of strong Huygens in dimension $1$.

The descended kernel and its value at the centre: for even $n=2k$ and data $(0,u_1)$, [F1] reads $u(x,t)=c^{1-n}{D}_t^{k-1}{W}_{u_1}(x,t)$ with the displayed $W_{u_1}$; for any $u_1$ supported in $B_{\epsilon }(x_0)$ with $0<\epsilon <c{t}_0$, choose $\delta >0$ with $c(t_0-\delta )>\epsilon$; then $(c^2{t}^2-{\rho }^2{)}^{-1\mathrm{/}2}$ is $C^{\mathrm{\infty }}$ on a neighbourhood of $[t_0-\delta ,t_0+\delta ]\times {\bar{B}}_{\epsilon }(x_0)$ on this compact parameter set, its derivatives times the bounded compactly supported $u_1$ have a constant integrable majorant, so [F4] lets $D_t^{k-1}$ be taken under the integral, giving $u(x_0,t_0)={\int }_{B_{\epsilon }(x_0)}{u}_1(y)K(\mathrm{\mid }y-x_0\mathrm{\mid },t_0)dy$ with $K$ as displayed; at $\rho =0$, [F6] with $\alpha =-1$ gives $K(0,t_0)=c^{1-n}(n!!V_n{)}^{-1}{c}^{-1}{D}_t^{k-1}[t^{-1}]=c^{-n}(n!!V_n{)}^{-1}(-1{)}^{k-1}(2k-3)!!t_0^{-(2k-1)}\mathrm{\ne }0$, and for $k=1$ the empty product is $1$, recovering Poisson's kernel at the centre [F2].

Nonzero value from interior data: $K(\rho ,t_0)$ is continuous in $\rho$ on a neighbourhood of $0$ because the differentiated expression is smooth there, and $K(0,t_0)\mathrm{\ne }0$, so there is $0<{\epsilon }_0<c{t}_0$ with $K$ of one constant sign on $[0,{\epsilon }_0]$; choosing now $0<\epsilon <{\epsilon }_0$ and any $u_1\in C_c^{k+1}(B_{\epsilon }(x_0))$ with $u_1\ge 0$, $u_1\equiv ̸0$ (take the translated smooth bump of [F8] with inner radius $\epsilon \mathrm{/}2$ and outer radius $\epsilon$), the product $u_1(y)K(\mathrm{\mid }y-x_0\mathrm{\mid },t_0)$ is continuous on the ball, of constant sign and nonzero somewhere, so by [F5] its integral $u(x_0,t_0)$ is nonzero; the data $(0,u_1)$ are admissible and supported in a compact subset of the open base ball, while the zero data give the value $0$, so strong Huygens fails in dimension $n$.

Conclusion: dimensions $1$ and every even $n\ge 2$ admit admissible data that vanish on a neighbourhood of the sphere $S(x_0,t_0)$ yet change the value $u(x_0,t_0)$, by [step 1.1] and [step 2.1]; by [F7] the strong Huygens principle holds in every odd dimension $n\ge 3$. Thus it fails in exactly dimensions $1$ and even $n\ge 2$, and the failures are witnessed by data supported strictly inside the base ball.