= Solution
Under the <Fourier transform>, the operator $A(t)=\partial_x^3+\phi(t)\partial_x$ is the <Fourier multiplier operator>
$$
\widehat{A(t)u}(\xi)
=i\bigl(\phi(t)\xi-\xi^3\bigr)\widehat u(\xi).
$$
Its symbol is purely imaginary because $\phi$ is real. Consequently $A(t)$ is <skew-adjoint> on $L^2(\mathbb R)$ with common domain $H^3(\mathbb R)$ and generates the <strongly continuous unitary group>
$$
\widehat{e^{rA(t)}u}(\xi)
=e^{ir(\phi(t)\xi-\xi^3)}\widehat u(\xi).
$$
The <Plancherel theorem> gives $\|e^{rA(t)}u\|_2=\|u\|_2$, so every $A(t)\in\mathcal G(1,0)$. Products of the frozen groups are also unitary; hence this is a <stable family of semigroup generators> with constants $1,0$.
For $u\in H^3(\mathbb R)$,
$$
\|(A(t)-A(s))u\|_2
=|\phi(t)-\phi(s)|\,\|\partial_xu\|_2
\leq|\phi(t)-\phi(s)|\,\|u\|_{H^3}.
$$
Since $\phi\in C^1(\mathbb R)$, the map $t\mapsto A(t)$ is continuously differentiable from $H^3$ to $L^2$. All hypotheses from part f are satisfied, so an <evolution family> exists on every finite interval $[0,T]$.
In this commuting <Fourier multiplier operator> example the solution operator can also be written explicitly:
$$
\boxed{
\widehat{U(t,s)u}(\xi)
=\exp\left(i\left[-(t-s)\xi^3
+\xi\int_s^t\phi(r)\,dr\right]\right)\widehat u(\xi)}.
$$
Its multiplier has absolute value one, directly confirming <strong continuity>, the <evolution family> law, preservation of $H^3$, and the required derivatives. The equation combines the dispersive <Airy equation> with a time-dependent <linear transport equation>.
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