Abstract. The paper considers the mixed two-phase Stefan problem for systems of reaction-diffusion equations. The behavior of free boundaries is studied

The existence result
Theorem. Suppose that the conditions of Theorem 2 and Lemma 1 are satisfied. Then there exists in a solution problems (1)-(5).
Proof. To prove the solvability of a nonlinear problem, one can use various theorems from the theory of nonlinear equations, remembering that the uniqueness theorem of the classical solution holds for it. We apply the Leray-Schauder principle [37], the established a priori estimates for all possible solutions of nonlinear problems and the solvability theorem in the Holder classes for linear problems. Moreover, the existence theorems for systems are the same as the theorem for the case of a single equation, since each of the equations can be considered as a linear equation for and with Holder-continuous coefficients.
Problem (1)-(5) is considered simultaneously with a one-parameter family of problems of the same type. The linear problem defines the transformation , to which the Leray-Schauder principle is applied. This operator is non-linear and depends on . Its fixed points with are solutions to the problem.
Denote by the Banach space of functions , on with norm , which satisfy the corresponding initial and boundary conditions of problem (10) and (11).
For each function and any number we denote by solutions to linear problems (A) and (B), the solutions of which exist and are unique; moreover, . Moreover, in the region D, as in Lemma 1, we pass to the parabolic equation with Holder-continuous coefficients in a fixed domain.
Uniform continuity and complete continuity of the transformation operator with respect to , uniform estimates for for solutions, and solvability of linear problems follows from the established a priori estimates of the Holder norms. The technique of proof is demonstrated in detail, for example, in (Chap. VII, [37]; Chap. VI, [36]).
This completes the proof of the theorem.