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

The Perona–Malik equation is an ill-posed forward–backward parabolic equation with some application in image processing. In this paper, we study the Perona–Malik type equation on a ball in an arbitrary dimension n and show that there exist infinitely many radial weak solutions to the homogeneous Neumann boundary problem for 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 an inhomogeneous partial differential inclusion problem, and to apply a Baire's category method to the differential inclusion to generate infinitely many solutions.