• forward-backward parabolic equation
• gamma-convergence
• gradient-flow
• maximal slope curves
• Perona-Malik equation
• semidiscrete scheme
• total variation flow

The Perona-Malik equation is a celebrated example of forward-backward diffusion process, introduced in the context of image processing.

A qualitative feature common to many approximation methods of this problem, confirmed by many numerical experiments, is the presence of three different time scales during the evolution, which have been called in recent literature ``fast time'', ``standard time'', and ``slow time''. In this thesis we study the ``slow time'' behavior of two different approximations of the Perona-Malik equation. Although the existence of these time scales seems independent from the approximation method, what exactly happens in the slow time from the quantitative point of view does depend on the approximation method.

First, we consider a mild regularization of this problem introduced in a recent paper. We prove that solutions of the regularized problem converge, in a slow time scale, to solutions of the total variation flow. The convergence result is global-in-time, and holds true in any space dimension.

The proof is based on the general idea that ``the limit of the gradient-flows is the gradient-flow of the limit functional''. To this end, we prove an abstract result for passing to the limit in the theory of maximal slope curves in metric spaces. It relates the Gamma-limit of a sequence of functionals to the limit of the corresponding maximal slope curves.

Then we consider the long time behavior of the semidiscrete scheme for the Perona-Malik equation in dimension one. We prove that the rescaled approximated solutions converge to solutions of a limit problem. This limit problem evolves piecewise constant functions by moving their plateaus in the vertical direction according to a system of ordinary differential equations. In this case, the collision of different plateaus makes the dynamic nontrivial, because the energies are not bounded from below. In order to overcome this difficulty, we prove a uniform older estimate up to the first collision time included and a well preparation result with a careful analysis of what happens at discrete level during collisions. Finally, we renormalize the functionals after each collision in order to have a nontrivial Gamma-limit for all times.