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

Theorem 2. Suppose that a function continuous in satisfies the conditions of problem (13)-(15) Assume that, for , and any , continuous functions and satisfy the condition
(16)
Under the conditions (16), the following estimate is valid
(17)
In addition, in the domain we have the estimate

And if it is also known that possesses in summable with a square generalized derivatives , , then
(18)
If , then the estimates (17)-(18) are also valid in ; where is a parabolic boundary, .
Proof . Since the estimates , , have been established, respectively, the boundedness of the function is obtained, by virtue of Theorem 3 in the paper [16], the inner estimates hold (17),(18).
If , then we redefine the function in the rectangle , by the formula . The function constructed in the rectangle (for which we retain the notation ) is continuous together with their derivative and everywhere, except for points of the line , satisfies an equation of the form (13) for which the conditions (14) and (15) are satisfied:
(19)
Now, applying Theorem 2 [16] to the problem (19), we obtain the estimate in the domain . Hence,

Similarly, we redefine the function in the rectangle by the formula .
Estimates in the domain , , give a general estimate of in .
Further, using Theorems 3 [16], we obtain the estimate
in .
The estimate for the highest derivative in is established using the results for the linear equations [17].
Theorem 3. Let the coefficients of equation
(20)
satisfy the Hölder conditions

Let the function be a solution of the parabolic equation (20), , . Then

where depends on
If , then there exists , such that