 Nonlinear diffusion equation

A nonlinear diffusion equation has a transport flux depending nonlinearly on the evolving field or its gradient. A common scalar conservative form isFor a nonnegative field in the porous medium equation $u_t=\Delta (u^m)$, $m>1$, the effective diffusion coefficient is $D(u)=m{u}^{m-1}$. It vanishes at zero density, allowing finite propagation in porous-medium diffusion. In logarithmic diffusion, $D(u)=\kappa /u$ for $u>0$, so low densities diffuse more strongly instead.

Gradient-dependent diffusion also belongs here: the Perona-Malik equation uses the flux $c(|\nabla u|)\nabla u$. Its highest-order coefficients depend on the derivative of that flux, so positive $c$ alone does not ensure that the equation is of parabolic type. Nonlinearity can therefore change propagation, smoothing and well-posedness, rather than merely changing a constant spreading rate.