1 Introduction

2 A priori estimates

In this section, we establish some a priori Schauder-type estimates necessary to prove the global solvability of the problem. To do this, we extensively use the maximum principle and the comparison theorems [?, ?, ?].

First of all, let us establish the boundedness of the functions , , . Further, a priori estimates are established for the highest derivatives.
Let be a solution of (1)-(6). Then
(7)
(8)
(9)
where , .
Proof. First we prove the positivity of the function . Take some arbitrary point such that . At this point, the right-hand side of (1) should be zero. And also at this point the function reaches its minimum value. Hence, according to the usual maximum principle for all and we obtain a contradiction regarding the condition . The resulting contradiction proves that in .
The proof of the positivity of is similar.
To estimate from above the functions , , we apply Theorem 7.2 of the paper [?] in the domain . Since the positiveness of the functions , are proved, then




and

i.e. the condition of the theorem is satisfied. Hence we have the estimates (7), (8).
Further, by virtue of positivity, and in and the boundary of conditions on the unknown boundary we have

Then from (6) we find .
In order to establish an upper bound for , in the problem (1)-(6) replacing

we get problems fo and

If we choose , , then by the maximum principle we have

From here
(10)
(11)

Therefore,

Thus, by the condition (6), the estimate (9) is established.
Theorem 2 is proved.
The boundary conditions of the problem (1) - (6) does not allow us to use the well-known results of [?]. Therefore, firstly, we introduce a transformation to straighten the free boundary

Then the domain corresponds to the domain , and the bounded functions , are a solution to the problem
(12)
where




Further, for each equation of the system, we separately formulate the corresponding problem:
(13)
(14)
where
Now, using the results of [?], we obtain the Hölder type estimates for systems of equations. Here and below, in relation to function spaces and the notation of norms in them, we will follow the notation of the work [?].
We introduce the notation
Let the function be continuous in together with and satisfy the conditions of the problem (13) in . And also, assume that . Then
(15)
If , then for
(16)
where , is a parabolic boundary. Proof. Since the estimates , , are established, and the boundedness of the function and is obtained, by virtue of Theorem 1 in the paper [?], the inner estimate (15) holds.
We now turn to the proof of (16). For these, by replacing the boundary conditions in the problem (13)

it is reduced to the homogeneous case. Then the problem (13) can be rewritten in the form
(17)
(18)
(19)
where
The coefficients of the equation of the problem (17) - (19) are bounded by Theorem 2.
First, we show an estimate up to the right bound for the function . Denote by , ; we continue the function to by the formula:

The new function in is continuous together with the derivative and everywhere, except for points of the line , satisfies problems of the form (17) - (19) for which
(20)
(21)
(22)
(23)
In this case, the properties of the coefficients of the problem (20) - (23) do not change. Using again the same scheme of Theorem 1 in [?], we find

Consequently,
(24)
Estimates like (24) are established in a similar way and in the area .
In the domain the replacement is

Estimates in the domain , , give a general estimate of in the closed domain .
Theorem 2 is completely proved.
Suppose that a function continuous in satisfies the conditions of problem (13). Assume that, continuous functions , for , , and arbitrary satisfy the conditions

In addition, if is in the area , , , then
(25)
And if it is also known that possesses in summable with a square generalized derivatives , , then
(26)
where , .
If , then the estimates (25), (26) are also valid in .
The proof of this theorem, as well as Theorems 3 and 4 in the paper [?]. Because we have information about the limited coefficients of the problem (13) and .
Note that arguing in the same way as above for the estimates (7)-(9), considering , we obtain estimates of the type (16), (26).
The estimates for the higher derivatives

are established with the use of the results obtained for linear equations [?, ?, ?].