An example of global classical solution for the Perona-Malik equation

TL;DR

This paper demonstrates that, unlike in one dimension, the higher-dimensional Perona-Malik equation admits smooth radial solutions with transcritical initial conditions, challenging previous assumptions about solution non-existence.

Contribution

It extends the understanding of the Perona-Malik equation by showing existence of global solutions in higher dimensions with transcritical initial data.

Findings

Existence of radial solutions in dimensions n >= 2

Counterexample to one-dimensional non-existence results

Solutions are of class C^{2,1} with transcritical initial conditions

Abstract

We consider the Cauchy problem for the Perona-Malik equation in an open subset of R^{n}, with Neumann boundary conditions. It is well known that in the one-dimensional case this problem does not admit any global C^{1} solution if the initial condition is transcritical, namely when the norm of the gradient of the initial condition is smaller than 1 in some region, and larger than 1 in some other region In this paper we show that this result cannot be extended to higher dimension. We show indeed that for n >= 2 the problem admits radial solutions of class C^{2,1} with a transcritical initial condition.