Nonlinear diffusion equations (porous-medium, fast-diffusion equation) and thin-film equations (with certain types of nonlinearities) can be recast as a gradient flows on an infinite-dimensional manifold.

The gradient-flow structure of these equations suggests a framework in which to study linear stability of selfsimilar solutions. Furthermore, in some cases it provides a way to show the asymptotic stability of the selfsimilar solutions, with optimal rates of convergence.

Of particular interest to us will be the stability of blow-up profiles of long-wave unstable thin-film equations. Future directions and open problems will also be discussed.