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

Theorem. 2. Let assume that are continuous in and satisfies (10) in together with (1). Then we have
(12)
Moreover, if the weak second derivatives belong to , then there exists , such that
(13)
Additionally, assume that, satisfying (10) in , is continuous with its derivatives and . Then
(14)
Proof. The estimates (12)-(14) for are immediate consequences of the results of [33].
In the case of problem (11), a priori estimates are constructed as follows. Estimates in the interior of the domain are established as in the case of problem (11). Further, replacing , we straighten out the boundary. Then domain is mapped to domain and for the function , we obtain an equation with bounded coefficients and the right-hand side. By the results of [35], we establish estimates for up to the right boundary. Estimates for the highest derivatives are obtained from the results for linear equations [36].
The uniqueness
First we derive an integral expression for the free boundary. To this end, we rewrite first equation of (2) as
(15)
where
Integrating (2) over , we obtain

Using (3)-(5), we have
(16)
Equation (16) will be the fundamental relation used in the uniqueness proof.
Theorem. The solution of the problem (1)-(5) is unique.
Proof. Let , and be one solution of (1)-(5), and let , and be another. Let Then each pair satisfies the identity (16). Subtracting, we obtain that

(17)
where are the solution between and .
Further, it is necessary to estimate the differences

For the function from the problem (10), we obtain
(18)
where the coefficients are bounded functions, i.e.
By the maximum principle [37], it follows from problem (18) that
(19)
where depends on .
By virtue of the established estimates for , , and , we get


where .
For the function from the problem (11), we find
(20)
where are bounded and continuous functions.
Here, the coefficients of the equation are continuous and bounded functions.
From this problem, invoking the maximum principle, we conclude that
(21)
where depends on and the coefficient .

Let Then

From (19) and (21), we have


where

Now that all the necessary estimates are established, applying the idea of ([1], Theorem 2) can complete the proof of the theorem.