Parabolic Equations in One Dimension: Thin Film, Kuramoto-Sivashinsky, and Magma Models

We first deal with a class of quasilinear fourth and higher-order operators, which, since the 1980s, began to play a determining role in the theory of nonlinear degenerate parabolic PDEs. It is not an exaggeration to say that nowadays higher-order nonlinear degenerate thin film equations are of the same fundamental importance as the second-order porous medium equations used in the 1950s-80s.