Proposition: Existence, uniqueness, and smooth parameter dependence

Equations (eq:Y1)–(eq:Y2) construct unique smooth decaying rod coefficients on the frozen classical domain. On compact time intervals their spatial decay is $O(|\mathbf{x}{|}^{-2})$. The coefficient maps are continuous in the smooth topology on fixed compact spatial supports.

Proof. The tensors $B_1,B_2$ are smooth and compactly supported in space. The first source is a divergence of a compact smooth vector. Convolution with the locally integrable fundamental solution (eq:newton) is smooth; derivatives can be transferred to the compact source.

The vector inside the second divergence is also compact: the possibly noncompact $\partial Y_1$ is multiplied by $B_1$. The same argument constructs $Y_2$.

For a compact smooth vector $b^i$,

Outside a sufficiently large ball the derivative of $1/|\mathbf{x}-\mathbf{y}|$ is $O(|\mathbf{x}{|}^{-2})$, uniformly for $\mathbf{y}$ in the source support. This proves the decay and its compact-time uniformity.

A difference of two first-order solutions is an entire harmonic function tending to zero at infinity. The maximum principle on large balls gives zero. Once $Y_1$ is fixed, the same argument proves uniqueness of $Y_2$.

On compact spatial sets, convolution and transferred derivatives are bounded by finitely many source seminorms. Outside such sets the explicit kernel estimates give the stated uniform bounds. The sources are finite smooth algebraic expressions in the input coefficients and the previously constructed $Y_1$. This proves continuity and smooth time dependence.