*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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