Principal diffusion coefficients ($c(s),c(s)+sc\prime (s)$)

For flux $c(|\nabla u|)\nabla u$, the derivative matrix is $c(s)I+c^{\prime }(s)\nabla u\otimes \nabla u/s$ with $s=|\nabla u|>0$. Its tangential and normal coefficients are $c(s)$ and $c(s)+s{c}^{\prime }(s)$. Positivity of the scalar conductance alone therefore does not guarantee forward parabolicity.