Suppose $X:\mathcal S\to\mathbb R^3$ is an embedding of a surface $\mathcal S\subset\mathbb R^3$. Then, the mean curvature normal of $\mathcal S$ is the Laplacian applied to the coordinates $\Delta X$, where $\Delta$ is the Laplace-Beltrami operator of $\mathcal S$. Hence, mean curvature flow can be written $\frac{\partial X}{\partial t}=-\Delta X,$ where we should note $\Delta$ is a function of $\mathcal S.$
This formula has an obvious connection to heat diffusion, where a function $u:\mathcal S\to\mathbb R$ diffuses along $\mathcal S$ (whose geometry stays fixed) via the PDE $\frac{\partial u}{\partial t}=-\Delta u$. In words, for an infinitesimal period of time, mean curvature flow diffuses the coordinate function along $\mathcal S$.
Suppose we have a more general second-order parabolic PDE of the form $$\frac{\partial u}{\partial t}=F(u, \nabla u, \Delta u).$$ Following the logic above, is there a class of geometric flows resembling this PDE? In particular, what can we say about geometric flows that can be written $$\frac{\partial X}{\partial t}=F(X, \nabla X, \Delta X)?$$
Are there examples of nonlinear parabolic PDE on $\mathcal S$ that translate to geometric flows through this formula? Unlike the counterpart involving $u$, here $\Delta$ would be itself a function of $X$.