= Solution
Solve the <adjoint transport equation>
$$
\varphi_t+x\varphi_x+\varphi=\psi
$$
backward with terminal value zero. Along the <characteristic flow map> $Z_{t,s}(x)=e^{s-t}x$, the required solution is
$$
\varphi(t,x)
=-\int_t^\infty e^{s-t}\psi\bigl(s,e^{s-t}x\bigr)\,ds.
$$
Differentiation under the integral verifies the equation. If $\psi$ has <compact support> in $[0,T]\times[-R,R]$, then $\varphi$ vanishes for $t>T$ and for $|x|>R$, so $\varphi\in C_c^1$ as required.
When the initial datum is zero, inserting this $\varphi$ into the <weak formulation> gives
$$
\int_0^\infty\!\int_{\mathbb R}u\psi\,dx\,dt=0
$$
for every $\psi\in C_c^1$. Thus $u=0$ <almost everywhere>. The difference of two bounded weak solutions has zero initial datum, so this proves uniqueness.
Solved by gpt-5.6-sol high.
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