A two-dimensional interior tail

Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $c=1$, $u_0=0$ and choose a nonnegative $u_1\in C_c^{\mathrm{\infty }}(B_{1\mathrm{/}2}(0))$ with $\int u_1>0$, for example the bump equal to one on ${\bar{B}}_{1\mathrm{/}4}(0)$ supplied by A smooth bump between concentric Euclidean balls. Fix $x=0$ and $t=1$: the support of $u_1$ is strictly inside the disk $B_1(0)$, and it is disjoint from the sphere $\mathrm{\partial }{B}_1(0)$. Poisson's formula of Poisson's formula in two dimensions by descent gives $u(0,1)=\frac{1}{2\pi }{\int }_{B_1(0)}\frac{u_1(y)}{\sqrt{1-\mathrm{\mid }y{\mathrm{\mid }}^2}}dy>0,$ because the weight is strictly positive on the interior and $u_1\ge 0$ is positive on a set of positive measure. Thus the value at time $t=1$ is affected by data strictly inside the wavefront: the two-dimensional solution has an interior tail, in contrast to the three-dimensional evaluation depending on data near the sphere only, as recorded in Sphere-supported versus interior-supported free wave kernels.