Abstract. The paper considers the mixed two-phase Stefan problem for systems of reaction-diffusion equations. The behavior of free boundaries is studied

Formulation of the problem statement.
We need to find the functions , and in the domains
(1)
(2)
(3)
(4)
(5)
where is a moving (unknown) boundary.
Problem (1)-(5) governs the dynamics of two species ( and ) over a bounded spatial domain, where the function (respectively ) stands for the population of the prey(respectively predator). The condition on conveys that initially occupies only a sub region of the whole domain. The Stefan condition, states that the speed at which the free boundary expands is proportional to the population-gradient at this location.
Regarding these tasks, the following conditions are assumed to be satisfied:
i. Parameters , , , , , and ( ) are positive constants;
ii. satisfy the following conditions:


The system (1)-(5) was studied in [25] for case. The techniques we employ in this study differ from the ones used in [25], and moreover, we prove a theorem on the uniqueness of the solution, which is of interest in its own right.
A priori estimates and global existence
First we establish some a priori estimates for the problem (1)-(5).
Lemma. 1. Let be a solution of problem (1)-(5) for . Then we have the following estimates
(6)
(7)
(8)
where .
Proof. First we prove the positivity of the function . Take an arbitrary point such that . At this point, the right-hand side of (2) 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 for . Therefore, $ $
Thus, by the strict maximum principle,
It then follows from the free boundary condition in (5) that for
It follows from the comparison principle that

where the solution of the problem

Similarly, considering the following initial value problem

by comparison principle we obtain that

To derive an upper bound for , to this end, we compare the auxiliary function defined by
(9)
We find that

By applying the maximum principle one more time, we obtain Then, (9) also implies that

Therefore, or
Then we get (8) which completes the proof.
We will establish Holder norm bounds and in and .
For each equation of the system (1)-(5), we formulate the corresponding problem
(10)
(11)
where
Under the condition (ii), without loss of generality, we can assume that , .