 Solution

The gas pressure exceeds the distant liquid pressure by the capillary pressure $\gamma /a$ associated with each cylindrical bubble. In the flat film the interface curvature is nearly zero, so its liquid pressure is close to the gas pressure and therefore exceeds the external liquid pressure by approximately $\gamma /a$. This pressure excess drives liquid out through the two transition regions.

Inside the flat region $h=h_0(t)$, so $h_{xxx}=0$. The extensional-force equation gives $(h_0{u}_x{)}_x=0$, hence $u_x$ is independent of $x$. Symmetry gives $u(0,t)=0$ and define $u(L,t)=U(t)$, soThe mass-conservation equation becomesIt contains no $x$ dependence, so an initially uniform film remains uniform within the flat region.