As for the inward melting inside an isothermal horizontal cylinder, where the unmelted solid core is constrained from moving under gravity, two drastically different melting patterns have been reported in the literature; the shape of the melting surface in the bottom portion has been observed to be either concave or convex. The present work was motivated to elucidate the main cause of these conflicting results. This was done first by obtaining the full transient solutions conforming to the model equations, and then by introducing the bifurcation times as well as a pair of bifurcating solutions. It was found that the convex pear-like melting surface could be well explained in terms of the primary bifurcating solution. Another mode of melting, in which the concave crater shape of the melting surface was observed. turned out to be also a kind of branching solution to the melting process of interest.