= Separable draining profile for power-law diffusion
For $\phi h_t=D_m(h^m h_x)_x$ on $0<x<L$, with $h(0,t)=0$ and zero right-hand <volume flux>, a <separation of variables> gives
$$
h=\left(\frac{\phi L^2}{mD_m\tau}\right)^{1/m}F(x/L),\qquad \tau=t+t_0,
$$
where the positive profile satisfies
$$
(F^mF')'+F=0,\qquad F(0)=0,\qquad F'(1)=0.
$$
Its boundary <volume flux per unit width> is
$$
Q=\frac{D_m}{L}\left(\frac{\phi L^2}{mD_m\tau}\right)^{(m+1)/m}c_m,\qquad c_m=\lim_{\xi\downarrow0}F^mF'=\int_0^1F\,d\xi.
$$
These profiles are exact separated solutions and describe the leading long-time discharge for a broad class of positive initial data. The virtual origin $t_0$ depends on the initial profile; it does not make the separated solution an exact representation of every initial condition.
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