Abstract
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.
| Original language | English |
|---|---|
| Pages (from-to) | 1889-1932 |
| Number of pages | 44 |
| Journal | Journal of Differential Equations |
| Volume | 258 |
| Issue number | 6 |
| DOIs | |
| State | Published - 15 Mar 2015 |
Keywords
- Baire's category method
- Infinitely many radial weak solutions in all dimensions
- Partial differential inclusion
- Perona-Malik type equation