Solution to a Two-Dimensional Nonlinear Parabolic Heat Equation Subject to a Boundary Condition Specified on a Moving Manifold

This paper is devoted to the study of a degenerating parabolic heat equation with nonlinearities of a general type in the presence of a source (sink) in the case of two spatial variables. The problem of initiating a heat wave propagating over a cold (zero) background with a finite velocity and a boundary condition specified on a moving manifold—a closed line—is considered. For this problem, a new existence and uniqueness theorem is proved, a numerical algorithm for constructing a solution based on the boundary element method, collocation method, and difference time approximation is proposed; a special change of variables of the hodograph-type transformation is used. New exact solutions to this equation in the case of power nonlinearities are found. A numerical algorithm is implemented, and a numerical experiment is carried out. A comparison of the constructed numerical solutions with exact ones (found both in this paper and earlier) showed good agreement. The numerical convergence in the time step and number of collocation points is proved.