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

3. UNIQUENESS AND EXISTENCE OF SOLUTION
To prove the uniqueness of the solution, we use the ideas of [19], [18].
We derive the integral representation equivalent to (3). To this end, we rewrite (3) as
(21)
where .
Integrating equation (21) over the domain we find

We get
(22)
Theorem 4. Under assumption of Theorems 1 and 2, problem (3)-(7) has a unique solution.
Proof. Let and are solutions of the problem (3)-(7) and, moreover,

Then, taking into account (22), we have
(23)
where is the solution defined between and .
From Theorem 1 we have that
(24)
Let . Then for we get an equation with bounded coefficients and the problem

where , , .
Hence by the maximum principle, we have

Due to the boundedness of the functions , , , we estimate the components of (23):






Let . Then

where .
Analogously we get
(25)
Next, we dividing (25) by we end up with
(26)
Now we estimate the integral term


Consider the auxiliary problem

We introduce the function

We have

Hence, by the principle of maximum

As

that

Consequently,

Since the right-hand side of (26) tends to zero when , then for sufficiently small we come to a contradiction. Consequently, , and further for .
The uniqueness of the solution of the problem for any is established as follows.
Let . If , then the issue will be resolved. Otherwise, assuming that the parameter is bounded and, repeating the calculations performed above in the interval , we again arrive at a contradiction.