Perona-Malik equation ($u_t=\mathrm{div}(c(|\nabla u|)\nabla u)$)

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 $c(r)+r{c}^{\prime }(r)$ normal to it. A negative normal coefficient allows formal sharpening but causes forward-backward ill-posedness. Positivity of $c$ alone is insufficient for parabolicity.