*Preprint*

**Inserted:** 13 nov 2006

**Year:** 2006

**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 $u:[-1,1]\times[0,T]\to\re$ is dense in $C^{1}([-1,1])$. Here ``classical solution'' means that $u$, $u_{t}$, $u_{x}$ and $u_{xx}$ are continuous functions in $[-1,1]\times[0,T]$.

**Keywords:**
Perona-Malik equation, classical solution, forward-backward equation, anisotropic diffusion

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