Some recent results concerning a class of forward-backward parabolic equations

Abstract

We study a quasilinear parabolic equation of forward-backward type in one space dimension, under assumptions on the nonlinearity satisfied by important mathematical models (e.g., the one-dimensional Perona–Malik equation). We first address a degenerate pseudoparabolic regularization of the equation, which takes time delay e¤ects into account, proving existence and uniqueness of positive solutions of the regularized problem in a space of Radon measures, as well as qualitative properties of such solutions. Then we address the vanishing viscosity limit of the regularized problem, proving that the limiting points (in a suitable topology) of the family of solutions of the regularized problem can be regarded as solutions of the original problem. In particular, we characterize the disintegration of the narrow limit of the family of Young measures associated with the absolutely continuous part of the solutions of the regularized problem, proving that it is a superposition of two Dirac masses with support on the branches of the graph of the nonlinearity φ. In the above study, the existence of a family of infinitely many entropy inequalities plays an important role.