Radial weak solutions for the Perona-Malik equation as a differential inclusion

Abstract

The Perona-Malik equation is an ill-posed forward-backward parabolic equation with major application in image processing. In this paper we study the Perona-Malik type equation and show that, in all dimensions, there exist infinitely many radial weak solutions to the homogeneous Neumann boundary problem for any smooth nonconstant radially symmetric initial data. Our approach is to reformulate the $n$-dimensional equation into a one-dimensional equation, to convert the one-dimensional problem into a differential inclusion problem, and to apply a Baire's category method to generate infinitely many solutions.