An investigation of two-dimensional, unsteady heat conduction in a solid with phase change at an exposed boundary has been conducted, including a perturbation analysis and numerical computations. The zero-order solution from which the perturbation analysis was developed is the steady, one-dimensional recession of a solid slab in response to a uniform heat input. The analysis has been carried out for small values of ( δ/ b) 2, where δ is the characteristic length for the temperature distribution in the solid, and b is the length scale of the lateral nonuniformity of the heat flux. A finite-difference analog of the governing equations was programmed, for the purpose of providing results for large nonuniformity in both heat flux and surface shape. Sample results are presented for two types of heat inputs: (a) one in which the spatial variation was sinusoidal, and (b) one which consisted of two constant levels connected by a half-wave cosine transition. Good agreement was obtained between predictions of the first-order analysis and results of the numerical computation for small times. For relatively large values of time, the computed surface shapes were found to become self-preserving. In case (a) groove shapes and depths tended to become time-independent, and in case (b), a straight ramp of constant slope tended to form. The mass loss of the solid was found to be smaller when the slab was subject to spatially uniform heating than when it was heated uniformly with the same average intensity (when lateral conduction was not neglected). For a given heating, the mass loss with lateral conduction was always less than when lateral conduction was neglected. An empirical correlation of the numerical results shows surface slopes or grooves depths to be proportional to a length scale which is associated with the nonuniformity in the heat flux, and its magnitude. The characteristic time for the appearance of surface features is ( δb)/ α., where α is the thermal diffusivity of the solid.