 Solution

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 $4h^{\ast }/(3\alpha )$.

Solved by gpt-5.6-sol high.