Existence and Uniqueness for Similarity Solutions of One Dimensional Multi-Phase Stefan Problems
DW Wilson · SIAM Journal on Applied Mathematics · 1978
The similarity solution of a multi-phase one-dimensional Stefan problem gives rise to a system of nonlinear equations for the coefficients of $\sqrt t $ which determine the speeds of the several interphase boundaries. We establish the existence and uniqueness of a solution to this system provided that a certain reasonable order relation exists among the boundary values. In addition, we establish a nonexistence result when most of our relations are satisfied but one is seriously violated. This implies the existence of physically plausible Stefan problems with constant boundary values for which there do not exist similarity solutions.