Theorem: A positive second-order descended evaluation

For $m>0$, the sourced family contains a tuple and a nonnegative compact test $f_s$ such that

Proof. Choose $F\in {C_c}^{\infty }({\mathbb{R}}^3)$, $F\ge 0$, equal to a constant $A>0$ near zero, and prescribe

Then

The nonnegative nonzero source gives $a(0)=-2\mathcal{N}\rho (0)>0$. Trace reversal yields

Since $h=0$ and $D(j)=0$, the coordinate equations give $w=v=0$. Hence the second relational density is ${\varphi }_2$. At the origin, the spatial Hessian of $F$ is zero and ${{\partial }_t}^2{\varphi }_0(0,0)=-m^{2}A$. Therefore

The zero initial data of ${\varphi }_2$ and its equation imply

For a sufficiently small positive time, ${\varphi }_2$ is positive on an open neighborhood, while ${\varphi }_0>0$ there. Choose a nonzero nonnegative $f_s$ in that neighborhood. The three assertions follow.