$\bullet$ *Physical model:* 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 has the form
$$\frac{1}{v_0}\frac{\partial}{\partial t}u=-\lambda^{3-m}\nabla\cdot(u^m\nabla\Delta u).$$
The parameter $v_0$ is a characteristic velocity of the interface, while the parameter $\lambda$ is the slip length at the interface. 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.)

$\bullet$ *Well-posedness:* At the end points $m=0$ and $m=3$ the solution is known to be singular (infinite dissipation for $m=3$, infinite interface velocity for $m=0$). Regularity of the solution in the interval $m\in(0,14/5)$ is proven in <A HREF="https://link.springer.com/content/pdf/10.1007%2Fs00205-015-0882-x.pdf">"Well-Posedness for a Class of Thin-Film Equations with General Mobility in the Regime of Partial Wetting."</A>

$\bullet$ *Conserved quantities*: Conservation of mass dictates that $\int u(x,t)dx$ should be $t$-independent. This implies that the self-similar solution
$$u(x,t)=t^{-\alpha}f(xt^{-\beta},\;\;m\alpha+4\beta=1,$$
must have $\alpha=\beta=1/(m+4)$. The corresponding self-similar solution is constructed in <A HREF="http://www.cs.vu.nl/~jhulshof/PAPERS/48.pdf">"Some aspects of the thin film equation."</A>