A new well-posed nonlinear nonlocal diffusion

A new nonlinear diffusion is proposed and analyzed. It is characterized by a nonlocal dependence in the diffusivity which manifests itself through the presence of a fractional power of the Laplacian. The equation is related to the well-known and ill-posed Perona–Malik equation of image processing. It shares with the latter some of its most cherished features while being well-posed. Local and global well-posedness results are presented along with numerical experiments which illustrate its interesting dynamical behavior mainly due to the presence of a class of metastable non-trivial equilibria.