= Solution
For every real $r$ and integer $n\geq1$, $|\sin r|\leq\min(1,|r|)$ gives
$$
r\sin^n r\leq |r|\,|\sin r|^n\leq |r|\,|\sin r|\leq r^2.
$$
This estimate covers both even and odd $n$, without any assumption on the sign of $u$. Since $a(t)\geq0$, the <energy estimate> from part (i) becomes
$$
E'(t)\leq2a(t)E(t),\qquad E(0)=0.
$$
Equivalently, an <integrating factor> gives
$$
\frac d{dt}\left[e^{-2\int_0^t a(s)\,ds}E(t)\right]\leq0.
$$
The bracket is nonnegative and starts at zero, so it vanishes identically. This is also the zero-data case of the <Gronwall inequality>. The vanishing <L2 norm> and continuity of $u$ imply
$$
\boxed{u(x,t)\equiv0.}
$$
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