The escaping-source control survives

For a nonzero nonnegative $\rho \in {C_c}^{\infty }((1,2))$, put $\eta (\mathbf{Y})=Y_1\rho (|\mathbf{Y}|)$. Let $\tau \in {C_c}^{\infty }(I)$, and take

These inputs vanish eventually on every fixed compact set, but

converges locally to a nonzero constant spatial vector times $\tau (t)$. Its first component is

Each finite input has the decaying original rod.

For the new weighted norm with $q>1$, the source norm grows like $R^{q-1}$. Thus the example is correctly excluded by the uniform weighted control needed for (eq:cutoffestimate). Local convergence alone remains insufficient.