A class of local classical solutions for the one-dimensional Perona-Malik equation

Abstract:

We consider the Cauchy problem for the one-dimensional Perona-Malik equation \[ u_{t}=\frac {1-u_{x}^{2}}{(1+u_{x}^{2})^{2}} u_{xx}\] in the interval $[-1,1]$, with homogeneous Neumann boundary conditions.

We prove that the set of initial data for which this equation has a local-in-time classical solution $u:[-1,1]\times [0,T]\to \mathbb {R}$ is dense in $C^{1}([-1,1])$. Here “classical solution” means that $u$, $u_{t}$, $u_{x}$ and $u_{xx}$ are continuous functions in $[-1,1]\times [0,T]$.