A semilinear parabolic system with a free boundary

(6)
(7)
where
,

Now, using the results of \cite{7}, we obtain Ho'lder-type estimates for systems of equations.
We introduce the notation
We formulate a theorem for the function .
Similar results are valid for .
Theorem 2. Let the function be continuous in together with and satisfy the conditions of the problem (6) in . Then
(8)
If , then for
(9)
where , parabolic boundary.


Proof. Since the estimates of (8)-(9), respectively, we obtain the boundedness of the function and , then, by Theorem Theorem of \cite{5} the internal estimate (8) holds.
By replacing

in the problem (6), the initial condition reduces to a homogeneous case. Then the problem (6) can be rewritten as
(10)
where
The coefficients of the equation of the problem (10) are bounded by virtue of the Theorem 1.
Further, the proof is carried out as in \cite{7}
The Theorem 2 is completely proved.
We proceed to obtaining a priori estimates of higher derivatives. From (2), we rewrite the equation as (7).
Theorem 3. Let the function be continuous in in the task with and satisfy the conditions of the problem (7), as well as

Then in , . And if it is also known that the functions in are summable with a square, generalized derivatives of and , then there is also , which

Theorem 3 is proved as Theorem 4 in (\cite{7}, Ch. III).
Estimates of higher derivatives are established using the results for linear equations \cite{5,8}.
Theorem 4. Let the coefficients of equation
(11)
satisfy the Holder conditions