= Solution
Along a <characteristic>, $\dot z=(1+t)^{-2}$ and $\dot H=0$. Integration gives $z+1/(1+t)=\xi$, so $\boxed{H=F(\xi)}$. A <characteristic> entering through $z=z_0$ at time $\tau$ has $\xi=z_0+1/(1+\tau)$ and carries $H=C_0(1+\tau)^{-4/3}$. Thus the boundary-fed and initially dry portions give
$$
\boxed{F(\xi)=\begin{cases}C_0(\xi-z_0)^{4/3},&z_0<\xi\leq z_0+1,\\0,&\xi>z_0+1.\end{cases}}
$$
Values at $\xi\leq z_0$ are never sampled in the finite-time physical region and need not be prescribed. The moving coating front is $z=z_0+t/(1+t)$. Explicitly,
$$
\boxed{H(z,t)=\begin{cases}C_0\left(z-z_0+\dfrac1{1+t}\right)^{4/3},&0\leq z-z_0\leq\dfrac{t}{1+t},\\0,&z-z_0>\dfrac{t}{1+t}.\end{cases}}
$$
The incompatible initial corner creates a jump at the leading coating front; a trace convention at the jump does not affect the transported volume. The wall's total stroke is $\int_0^\infty v_0\,dt=1$, so within this leading transport model the final profile is
$$
\boxed{H_\infty(z)=\begin{cases}C_0(z-z_0)^{4/3},&0<z-z_0<1,\\0,&z-z_0>1.\end{cases}}
$$
Earlier-deposited fluid is thicker and sits farther up the finite coated interval. Its total scaled volume is $3C_0/7$, agreeing with $\int_0^\infty v_0H(z_0,t)\,dt=C_0\int_0^\infty(1+t)^{-10/3}dt$. This is a <finite-stroke coating profile> for the leading matched model.
\Image[/past-exam-of-the-topics-in-fluid-mechanics-course-of-the-university-of-oxford/2009-blink-film-profile.png]
{title=Transported coating profiles during a finite wall stroke}
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