A semilinear parabolic system with a free boundary

A priori estimates
In this section, we establish some a priori estimates of the Schauder type, which will be used to prove the global solvability of the problem. Moreover, the maximum principle and comparison theorems \cite{3} are widely used.
The main difficulty in constructing a nonlocal theory of nonlinear problems is to obtain estimates of the first derivatives with respect to the spatial variable. There are various methods for obtaining a priori estimates.
In this paper, we will use the method for obtaining a priori estimates proposed in \cite{3,4}. Therefore, we adhere to the notation adopted by \cite{4, 14, 17}.
Theorem 1. Let functions are a solution to the problem (1) -(5). Then the inequalities



where ,
Proof. First we prove the positivity of the function . Take an 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 arrive at a contradiction. The resulting contradiction proves that in .
Similarly, we have in . Since , Hopf lemma then implies that for all . It then follows from the free boundary condition (5) that for .
We will employ the result of Theorem 5.1 ([16], Chapter 2). If we take then satisfies all the assumptions of the theorem .
Consequently,

To set the upper bound for in the (1) task, replacing

we get



due to the choice of and by the maximum principle, we have From here . Consequently or
A similar argument using the property of the lower solution gives

Then, from the Stefan condition, the estimate (5) is obtained.

Theorem 1 is proved.
Before proceeding to the establishment of a priori estimates, we reduce the boundary conditions at fixed boundaries to zero. This allows us to take advantage of the results of \cite{3, 4, 7}.
Now in the new task (1) -(5) we will replace the independent variables

Then the region corresponds to the region and limited features , are the solution to the problem