Porous medium equation ($u_t=\Delta (u^m)$)

The porous medium equation $u_t=\Delta (u^m)$ with $m>1$ is a nonlinear partial differential equation whose diffusivity vanishes at $u=0$. Unlike the heat equation, it has compactly supported solutions with finite propagation speed.

For $C_t=k\nabla \cdot (C\nabla C)$ in three dimensions and total mass $4\pi M$, the nonnegative radial source solution is $C=M^{2/5}(kt{)}^{-3/5}f(r/(Mkt{)}^{1/5})$, with $f(\xi )=(75^{2/5}-{\xi }^2{)}_{+}/10$. Integrating the similarity equation once gives $f{f}^{\prime }=-\xi f/5$, and normalization ${\int }_0^{\infty }{\xi }^{2}fd\xi =1$ fixes the compact-support radius. The flux vanishes at the moving front, so the profile solves the equation weakly and approaches the prescribed point mass as time tends to zero.

For the planar diffusion equation $n_t=D_0\nabla \cdot [(n/n_0{)}^2\nabla n]$, a mass-preserving radial similarity solution has $n=n_0{\lambda }^{-2}F(r/(r_0\lambda ))$. Substitution reduces the equation to $(\xi F^2{F}^{\prime }{)}^{\prime }=-({\xi }^{2}F{)}^{\prime }$ when ${\lambda }^5\dot{\lambda }=D_0/r_0^2$. Zero central flux and $F(0)=1$ give the compact profile shown above, with ${\lambda }^6=6D_{0}t/r_0^2$. Its height decays as $t^{-1/3}$ and its radius grows as $t^{1/6}$; its total mass is $2\pi n_0{r}_0^2/3$.

In a porous medium equation with exponent $m>1$, the effective diffusion coefficient vanishes at zero density. A nonnegative compactly supported source profile can therefore retain a finite front for every finite positive time. For example, the radial cubic-diffusion source profile has support radius proportional to $t^{1/6}$. This notion concerns a moving diffusion front; it does not require the constant characteristic cone used for a wave equation.

Consider a nonlinear diffusion equation with uniform supply, $\varphi h_t=D_m(h^m{h}_x{)}_x+R$, on $x>0$, with an absorbing boundary $h(0,t)=0$ and initially $h=0$. Far from the boundary, $h=Rt/\varphi$. Balancing the time derivative, supply, and diffusion givesThe similarity solution satisfiesThe outward boundary volume flux per unit width is $Q=R\ell c_m$, where $c_m={\lim }_{\eta \downarrow 0}{F}^m{F}^{\prime }$. Integrating the equation givesThus the growing region of depleted storage fixes the discharge prefactor, and $Q\propto t^{(m+1)/2}$.

For $\varphi h_t=D_m(h^m{h}_x{)}_x$ on $0<x<L$, with $h(0,t)=0$ and zero right-hand volume flux, a separation of variables giveswhere the positive profile satisfiesIts boundary volume flux per unit width isThese profiles are exact separated solutions and describe the leading long-time discharge for a broad class of positive initial data. The virtual origin $t_0$ depends on the initial profile; it does not make the separated solution an exact representation of every initial condition.

The Barenblatt solution is a mass-preserving similarity solution of the porous medium equation. For $u_t=\beta (u{u}_x{)}_x$ in one dimension it has the form

For the planar porous medium equation $n_t=[D_0/(3n_0^2)]\Delta (n^3)$, the mass-$Q$ point-source similarity solution is $n=n_0{\lambda }^{-2}\sqrt{[1-r^2/(r_0^2{\lambda }^2){]}_{+}}$, with $\lambda =(6D_{0}t/r_0^2{)}^{1/6}$ and $r_0^2=3Q/(2\pi n_0)$. Its front grows as $t^{1/6}$ and its central density falls as $t^{-1/3}$. The mass conservation integral uses the planar area element $2\pi rdr$. Although the density slope diverges at the front, its diffusion flux vanishes there, giving a compactly supported weak solution with initial measure $Q{\delta }^{(2)}$.

For $n,A>0$, the one-dimensional porous medium equation $h_t=A(h^n{h}_x{)}_x$ has a volume-preserving similarity solution $h=[c_n(R^2-x^2)/(At){]}_{+}^{1/n}$, where $c_n=n/[2(n+2)]$ and $R\propto t^{1/(n+2)}$. On a reflecting half-line with conserved area $M$, $R=[M(At/c_n{)}^{1/n}/I_n{]}^{n/(n+2)}$, where $I_n=\frac{1}{2}B(\frac{1}{2},1+1/n)$. For a whole symmetric release use half its total area as $M$. The Beta function fixes normalization, while finite-front zero volume flux fixes the edge condition.

The source-type similarity solution of $h_t=D(h^2{h}_x{)}_x$ with whole-line conserved area $\mathcal{V}$ is $h=H(t)[1-x^2/L(t{)}^2{]}_{+}^{1/2}$, where $L=2(\mathcal{V}/\pi {)}^{1/2}(Dt{)}^{1/4}$ and $H=(\mathcal{V}/\pi {)}^{1/2}(Dt{)}^{-1/4}$. Its volume flux vanishes at the finite front, while the velocity tends to $\dot{L}=L/(4t)$. A reflecting half-line uses twice its prescribed area in these formulas.

For the radially symmetric equationin two dimensions, the mass-$N$ similarity solution has the compactly supported formwhere mass normalization gives $x_0=(16/\pi {)}^{1/4}$. Its support radius grows as $t^{1/4}$ and its occupied area grows as $t^{1/2}$.