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

TL;DR

This paper investigates how subcritical regions evolve in solutions to the Perona-Malik equation across different dimensions and symmetries, revealing expansion behaviors and non-existence results.

Contribution

It provides a detailed analysis of subcritical region dynamics in the Perona-Malik equation for various cases, including explicit expansion rates and counterexamples.

Findings

Subcritical regions increase in one-dimensional solutions.

Radial solutions in higher dimensions expand and cover the domain in finite time.

Existence of solutions where subcritical regions do not expand.

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 increase, but there is no estimate on the expansion 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 non-existence 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 non-smooth free boundary problems.