= Solution
Let $z$ measure height above the sponge. Under the <long-wave approximation>, <hydrostatic pressure> gives $p_x=\rho gH_x$, and the horizontal balance for <viscous fluid flow> is
$$
\mu u_{zz}=\rho gH_x.
$$
The <no-slip boundary condition> $u(0)=0$ and <stress-free boundary condition> $u_z(H)=0$ give
$$
u=\frac{\rho g}{\mu}H_x
\left(\frac{z^2}{2}-Hz\right).
$$
Integrating this <lubrication theory> profile gives the horizontal <volume flux> per unit span
$$
q=\int_0^H u\,dz
=-AH^3H_x,
\qquad
A=\frac{\rho g}{3\mu}.
$$
Local <mass conservation> includes the downward loss found in part i:
$$
H_t+q_x=-DH^{1-\beta}.
$$
Consequently the required nonlinear <partial differential equation> is
$$
\boxed{H_t=A(H^3H_x)_x-DH^{1-\beta}}.
$$
If the imposed inlet flux is $Q_0t^\alpha$ and the moving front is $x=x_N(t)$, sufficient <boundary conditions> are
$$
-AH(0,t)^3H_x(0,t)=Q_0t^\alpha,
$$
$$
H(x_N(t),t)=0,
\qquad
q(x_N(t),t)=0.
$$
For a current released onto a dry substrate one also takes the <initial condition> $H(x,0)=0$ away from the source. The front position is part of this <moving-boundary problem>.
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