On the evolution of subcritical regions for the Perona–Malik equation

Abstract

The Perona–Malik equation is a celebrated example of forward-backward parabolic equation. The forward behavior takes place in the so-called subcritical region, in which the gradient of the solution is smaller than a fixed threshold. In this paper we show that this subcritical region evolves in a different way in the following three cases: dimension one, radial solutions in dimension greater than one, general solutions in dimension greater than one.
In the first case subcritical regions do not shrink, that is, that they expand with a nonnegative rate. In the second case they expand with a positive rate and always spread over the whole domain after a finite time, depending only on the (outer) radius of the domain. As a by-product, we obtain a nonexistence result for global-in-time classical radial solutions with large enough gradient. In the third case we show an example where subcritical regions do not expand.
Our proofs exploit comparison principles for suitable degenerate and nonsmooth free boundary problems.