= Solution
First neglect <inertia> relative to the $O(\epsilon)$ <pressure> correction, requiring $F\ll1$ for smooth fields on these scales. The positive <velocity> is
$$
u=A^{1/4}\bigl[1-\epsilon(\sqrt A)_x\bigr]^{1/2}=A^{1/4}-\frac\epsilon4A^{-1/4}A_x+O(\epsilon^2).
$$
Multiplying by <area> gives the <discharge> $Q=A^{5/4}-(\epsilon/4)A^{3/4}A_x+O(\epsilon^2)$. Hence
$$
\boxed{A_t+(A^{5/4})_x=\frac\epsilon4\bigl(A^{3/4}A_x\bigr)_x.}
$$
The leading equation is a nonlinear <kinematic wave> equation. The <pressure> correction has positive <diffusion coefficient> for $A>0$ and smooths steep gradients. Dropped terms are $O(\epsilon^2)$ and inertial terms $O(\epsilon F)$ on the assumed scales; sufficiently thin fronts require a separate balance rather than an unrestricted <Taylor expansion>.
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