TY - JOUR
T1 - Radial weak solutions for the Perona-Malik equation as a differential inclusion
AU - Kim, Seonghak
AU - Yan, Baisheng
N1 - Publisher Copyright:
© 2014 Elsevier Inc.
PY - 2015/3/15
Y1 - 2015/3/15
N2 - 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.
AB - 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.
KW - Baire's category method
KW - Infinitely many radial weak solutions in all dimensions
KW - Partial differential inclusion
KW - Perona-Malik type equation
UR - http://www.scopus.com/inward/record.url?scp=84921457783&partnerID=8YFLogxK
U2 - 10.1016/j.jde.2014.11.017
DO - 10.1016/j.jde.2014.11.017
M3 - Article
AN - SCOPUS:84921457783
SN - 0022-0396
VL - 258
SP - 1889
EP - 1932
JO - Journal of Differential Equations
JF - Journal of Differential Equations
IS - 6
ER -