Nonlinear diffusion equation

A nonlinear diffusion equation lets diffusivity depend on the evolving field; the porous-medium equation is a standard example.

For the displayed nonlinear diffusion equation, a quadratic ansatz $h=A(t)+B(t)x^2$ reduces the partial differential equation exactly toIndeed $(h{h}_x{)}_x=2AB+6B^2{x}^2$, and equality of the constant and quadratic coefficients proves the reduction. For nonzero curvature, integration gives $B=-1/[6(t-t_0)]$. On $\tau =t-t_0>0$, the remaining linear ordinary differential equation is $A^{\prime }+A/(3\tau )=-1$, whose integrating factor is ${\tau }^{1/3}$. Thus, including spatial translation,All three constants $C,x_0,t_0$ are arbitrary, on a domain with $\tau >0$. When the quadratic coefficient vanishes, the more general affine ansatz gives $h=px+(p^2-1)t+q$.

A narrower ansatz $h=a{t}^{-1/3}(1-x^2/t^{2/3})-bt$ fixes two coefficients together and permits exactly $(a,b)=(1/6,3/4)$ or $(0,1)$. The zero value of $a$ is a valid spatially constant branch, not an algebraic exception to discard. These finite-parameter families are exact solutions; they are not the complete general solution of the nonlinear partial differential equation, and their sums need not be solutions.

For a positive density $u$, logarithmic diffusion is the nonlinear diffusion equation $u_t=\kappa \Delta \log u$, equivalently $u_t=\kappa \mathrm{div}(u^{-1}\nabla u)$. Its diffusion coefficient is inversely proportional to the density. In one dimension, a mass-$M$ self-similar solution isIts integral is $M$, and substitution verifies the equation. This is $M$ times a Cauchy distribution density of scale $2\pi \kappa t/M$. It tends weakly to $M$ times the Dirac delta distribution as $t\downarrow 0$; the slow algebraic tails illustrate the enhanced diffusion where density is small.

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.

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.

For one-dimensional flux $F(p)=p|p|e^{-p^2/(2{\lambda }^2)}$, the nonlinear diffusion equation is $u_t=F^{\prime }(u_x)u_{xx}$. The sole one-dimensional member of the principal diffusion coefficients is positive for $0<|u_x|<\sqrt{2}\lambda$, negative above $\sqrt{2}\lambda$, and zero at the two endpoints. Negative diffusion gives formal sharpening with ill-posedness through high-frequency growth. The extra factor $|p|$ matters: the unweighted exponential conductance has threshold $\lambda$ instead.

Computing conductance from a smoothed gradient gives $u_t=\mathrm{div}(c(|\nabla (G_{\sigma }\ast u)|)\nabla u)$. The image edge detector is nonlocal, while the local highest-order diffusion has positive coefficient when conductance is positive. A smooth kernel, lower conductance bound and appropriate regularity/boundary hypotheses support a well-posed regularized filter.

For $C_t=k\nabla \cdot (C\nabla C)$ with total mass $2\pi M$,Its compact support has radius $r_0(t)=(32Mkt{)}^{1/4}$.

The substitutionreduces $C_t=k\nabla \cdot (C\nabla C)+aC$ to
$G_{\tau }=k\nabla \cdot (G\nabla G)$.