A quasilinear parabolic equation of the reaction-diffusion type with free boundary, M. S. Rasulov 1, and A. K. Norov 1

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.
Theorem1. Let , be a solution to problem (3)-(7).Then there exist positive constants , independent such that
(8)
(9)
Proof. First we prove the positivity of the function . Take some arbitrary point such that . At this point, the right-hand side of (3) 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 .
To get the upper bounds, we proceed as follows. We construct an auxiliary problem for an ordinary differential equation:
(10)
Its solution is given by the following explicit formula:
(11)
where , . Comparing with yields that [19]

Further, taking into account the condition (III) and the positivity of the function in the domain , we find . Hence, from (6) we get .
To set an upper bound for , in problem (3)-(6), replacing
(12)
and we get the problem with respect to

By choosing in , then we have . Therefore, taking into account (12), we find , which is equivalent to . The proof is complete.
Initial estimates have been received. Now, using the results of the work [16], we establish a priori estimates for and higher derivatives.
First, we obtain a priori estimates for . Due to the boundary conditions (3)-(7), we cannot use the results of the work [16]. Therefore, we introduce the following transformation

to straighten the free boundary. Then the domain is transformed into the domain , and the bounded function is a solution to the problem
(13)
(14)
(15)
where , , , .