# An example of global classical solution for the Perona-Malik equation

created by gobbino on 07 Jul 2009

[BibTeX]

Preprint

Inserted: 7 jul 2009

Year: 2009

Abstract:

We consider the Cauchy problem for the Perona-Malik equation $$u{t}=\mathrm{div}\left(\frac{\nabla u}{1+ \nabla u {2}}\right)$$ in an open set $\Omega\subseteq\re^{n}$, with Neumann boundary conditions. It is well known that in the one-dimensional case this problem does not admit any global $C^{1}$ solution if the initial condition $u_{0}$ is transcritical, namely when $\nabla u_{0}(x) -1$ is a sign changing function in $\Omega$. In this paper we show that this result cannot be extended to higher dimension. We show indeed that for $n\geq 2$ the problem admits radial solutions of class $C^{2,1}$ with a transcritical initial condition.