Q1: there is no physical model that gives this equation for arbitrary $m$; the values $m=1,2,3$ appear in viscous flow, as summarized in Viscous Thin Films: For the no-slip boundary condition one has $m=3$, different slip-boundary conditions give $m=2$ (Navier slip) or $m=1$ (Hele-Shaw cell).
Q2: this tutorial by J. Hulshof seems a good entry point to the mathematical literature (which considers arbitrary $m$).