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

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