On the structure of weak solutions of the Riemann problem for a degenerate nonlinear diffusion equation

An explicit form of weak solutions to the Riemann problem for a degenerate nonlinear parabolic equation with a piecewise constant diffusion coefficient is found. It is shown that the lines of phase transitions (free boundaries) correspond to the minimum point of some strictly convex and coercive function of a finite number of variables. A similar result is true for Stefan's problem. In the limit, when the number of phases tends to infinity, there arises a variational formulation of self-similar solutions to the equation with an arbitrary nonnegative diffusion function.