**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 <A HREF="https://www.mis.mpg.de/applan/research/viscousthinfilms.html">Viscous Thin Films:</A> 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).

Here is a way to understand from dimensional arguments the physics condition $m\leq 3$. Since $u$ has the dimension of length, the gradient $\nabla$ has dimension of 1/length, and the Laplacian $\Delta$ has dimension of 1/length$^2$, the thin-film equation in physical units of length $\lambda$ and time $\tau$ has the form
$$\tau\frac{\partial}{\partial t}u=-\lambda^{3-m}\nabla\cdot(u^m\nabla\Delta u).$$
The physical interpretation of the parameter $\lambda$ is the slip length. The no-slip boundary condition has $\lambda=0$, which enforces $m=3$. When there is slip we may have $m<3$, but not $m>3$, because then the right-hand-side would diverge in the limit $\lambda\rightarrow 0$, which is unphysical. (It would imply a divergent interface velocity.)

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