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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.

The equation is a conservation law $n_t+J_x=0$ with diffusive flux $J=-D(n_0/n)n_x$. For positive density, insects move down the density gradient, with effective diffusion coefficient $D{n}_0/n$: dispersal is stronger in low-density regions. This is logarithmic diffusion, since the equation is $n_t=D{n}_0(\log n{)}_{xx}$.

Put $\tau =Dt$ and $\xi =x/{\tau }^{\beta }$. A self-similar solution $n=n_0{\tau }^{-\beta }g(\xi )$ conserves total number when ${\int }_{\mathbb{R}}g=1$. Its width shrinks to zero as $t\downarrow 0$; for an integrable nonnegative $g$, changing variables in the integral against a bounded continuous function proves convergence to $n_0$ times the Dirac delta distribution at the origin.

Substitution givesMatching powers for a nonstationary profile forces $\beta =1$. The profile equation integrates to $g^{\prime }/g=-\xi g+C$. For a centred even profile, $C=0$, and integration of $g^{\prime }=-\xi g^2$ gives $g=2/({\xi }^2+a^2)$. Normalization requires $2\pi /a=1$, henceThis is $n_0$ times a Cauchy distribution density with scale $2\pi Dt$. It is positive, has total mass $n_0$, and directly satisfies the nonlinear diffusion equation for $t>0$.