A quasisteady multidimensional Stefan problem is treated by means of similarity, asymptotic methods, and numerical analysis. The continuous penetration of a solid “erodee” (a thick broad slab of ice) is effected by the gradual consumption of a solid “eroder” (composed of lithium). The ice erodee melts and the (liquid) water diffuses to the lithium eroder, where an exothermic, indefinitely rapid, diffusion-controlled surface reaction occurs. The products of the reaction convey some of the chemically-generated heat to the surface of the ice erodee. so that further melting may occur, and the process may proceed. The analysis seeks to estimate the quasisteady rates of consumption of the erodee and eroder, for the (convenient) quasi-selfsimilar case of confocal parabolas defining the boundaries or the fluid layer between the two solids. The analysis is simplified by observing that the flow is of lubrication-theory type: creeping, primarily unidimensional motion owing to the thinness of the fluid layer between the eroder and erodee, with diffusion of heat and mass predominantly in the direction perpendicular to the principal motion. Because the lithium eroder would float on a water layer, a weight must be added to the eroder to maintain close proximity of eroder and erodee. and the analysis is completed by consideration of the force balance on the eroder.