Q1: there is no physical model that gives this equation for arbitrary $m$; only $m=1,2,3$ appear in viscous flow (see the references by Zurab Silagadze); an instructive derivation for $m=3$ based on a variational principle is given in <A HREF="http://iopscience.iop.org/article/10.1088/0953-8984/27/8/085005">A variational approach to thin film hydrodynamics of binary mixtures</A> (2015).

Q2: this [tutorial](http://www.cs.vu.nl/~jhulshof/PAPERS/48.pdf) by J. Hulshof seems a good entry point to the literature.