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.

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• PM_GlobTrans.pdf