The Perona-Malik equation is a celebrated example of nonlinear forward-backward diffusion, introduced in the context of image denoising. It can be viewed as the formal gradient-flow of a functional with a convex-concave integrand. In spite of its mathematical ill-posedness, numerical experiments exhibit better than expected behavior of its solutions.
After a general introduction to the equation itself, we present a few approximation schemes, some classical and some more recent. The approximating solutions show distinct behavior on three time scales (called fast, standard, and slow time). We provide a rigorous explanation for the slow time behavior of the different approximations.
In order to carry out this analysis, we prove an abstract result about passing to the limit in gradient-flows (in the more general context of the theory of maximal slope curves in metric spaces). We are guided by the general principle that "the limit of a family of gradient-flows is the gradient-flow of the limiting functional".