On the surface of a semi-infinite solid, a point heat source releases a power ##q##; apart from this, the surface of the solid is adiabatic. The heat melts the solid so that a molten pool forms and grows. Let's hypothesize that the pool temperature is homogeneously equal to the melting temperature (hopefully this simplifies the problem) and the liquid and solid densities are the same. The melt pool exchanges heat with just the rest of the solid by conduction through its surface. A steady-state regime eventually is reached, in which q is equal to the heat flux through the surface of the pool. In this condition, what will the shape of the pool look like? Symmetry arguments suggest a spherical cap. What are its dimensions as function of q, density, thermal conductivity, melting temperature, and initial temperature of the solid? Is it possible to find the temperature field in the solid.