In this lecture we study the nonlinear stability of uniform
steady states of a fourth-order reaction diffusion equation of the
Swift-Hohenberg type. An important consequence of this stability is that
the equation preserves the sign of a set of its solutions provided some
appropriate restrictions on the parameters and the initial data are met.
This equation and related ones are also used in the contexts of
population dynamics, where obviously the solutions must be nonnegative
functions. They have a very rich dynamics including travelling waves,
periodic patterns and localised structures.