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When sliding dominates and the whole cap is below the snowline, its thickness obeysWith $\alpha =0$, this is a one-dimensional porous medium equation. Its mass-preserving Barenblatt solution isWithin its support,Now set $h=h_{ss}+ct$. Since $h_{ss}$ solves the unforced equation,Matching the uniform ablation term $-\alpha$ givesThe similarity solution has a virtual origin. Writing $s=t+t_0$ gives the exact decaying familyIf its initial centre thickness is $h^{\ast }$, then $C=(h^{\ast }+3\alpha t_0/4)t_0^{1/3}$ and the exact extinction time isIts value depends on the initial cap width through the virtual age $t_0$, but its dimensional scale is unambiguouslyThe additive melting correction alone has the corresponding time $4h^{\ast }/(3\alpha )$.

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 variable $w=h^2$ converts the shallow V-channel lubrication flux equation into a porous medium equation:Indeed $h^4{h}_x=w^{3/2}{w}_x/2=(w^{5/2}{)}_x/5$. Its fixed-mass source-type solution gives $h\propto t^{-1/7}$ and support length proportional to $t^{2/7}$. Integrating the similarity equation once gives $C{h}^4{h}_x=-2x{h}^2/(7t)$, and hence $h=[3(X(t{)}^2-x^2)/(7Ct){]}_{+}^{1/3}$. The constant-volume similarity in a V-shaped channel fixes $X$ from the positive integral $J={\int }_{-1}^1(1-s^2{)}^{2/3}ds$: $X=(\alpha V/J{)}^{3/7}(7Ct/3{)}^{2/7}$.

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 $Q{\delta }^{(2)}$.