Planar volume-conserving nonlinear-diffusion similarity ($h_t=A(h^n{h}_x{)}_x$)

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.