Manuscript XVIII-A@2 · XVIII-A-v2.tex · proposition · 0f4336ec

Proposition: Late source agreement does not determine the response

Preparation residuals and retarded response · lines 629–650 · proof included or linked

XVIII-A@2#prop:lateresponse, XVIII-A@2#eq:layermoments

There are smooth sources $s_{1}, s_{2}$ agreeing for every $t \geq ε$ whose zero-data retarded solutions

u_{i}^{''} = s_{i}, u_{i} (0) = u_{i}^{'} (0) = 0

differ for every $t > ε$ .

Proof. Take nonzero $a \geq 0$ in $C_{c}^{\infty} ((0, ε))$ and set $s_{1} = a$ , $s_{2} = 0$ . For $t \geq ε$ ,

u_{1} (t) - u_{2} (t) = t M_{0} - M_{1}, M_{0} = \int_{0}^{ε} a (r) d r, M_{1} = \int_{0}^{ε} r a (r) d r .

Here $M_{0} > 0$ and $M_{1} < ε M_{0}$ , so the difference is strictly positive for $t > ε$ .

Displayed formulas in this passage

XVIII-A@2#eq:layermoments · line 642
$u_{1} (t) - u_{2} (t) = t M_{0} - M_{1}, M_{0} = \int_{0}^{ε} a (r) d r, M_{1} = \int_{0}^{ε} r a (r) d r .$
unlabeled · line 633
$u_{i}^{''} = s_{i}, u_{i} (0) = u_{i}^{'} (0) = 0$