= Perona-Malik equation
{c}
{title2=$u_t=\operatorname{div}(c(|\nabla u|)\nabla u)$}
The Perona-Malik equation uses decreasing diffusivity to smooth low-gradient image regions while reducing transport across edges. For $r=|\nabla u|>0$, its principal diffusion coefficients are $c(r)$ tangent to a level set and $c(r)+rc'(r)$ normal to it. A negative normal coefficient allows formal sharpening but causes forward-backward <ill-posedness>. Positivity of $c$ alone is insufficient for parabolicity.
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