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

On a uniqueness of solution for a Reaction-diffusion type System with a Free Boundary
A. N. Elmurodov
E-mail: [e-mail]
V. I. Romanovskii Institute of Mathematics, Academy of Sciences of Uzbekistan, Universitetskaya street 4-B, Tashkent, 100174 Uzbekistan
Abstract. The paper considers the mixed two-phase Stefan problem for systems of reaction-diffusion equations. The behavior of free boundaries is studied. A priori estimates of Schauder type are established, on the basis of which the unique solvability of the problem is proved.
Keywords: Nonlinear problem, predator-prey model, parabolic system, a priori estimates, free boundary.
Introduction
Today, humanity faces increasingly serious environmental and epidemiological problems, such as environmental pollution, invasion of exotic species, the emergence of new infectious diseases and the resumption of existing epidemiological diseases. Recently, Mathematical modeling has been successfully used successfully to study many biomedical and epidemiological problems, and in all these contexts nonlinear and complex dynamics have been [1-14].
Over the past twenty years, significant progress has been made in the mathematical modeling of biomedical processes, which has led to the creation of more complex models consisting of systems of nonlinear partial differential equations.
There are many different biological considerations regarding the diffusion competition problem associated with (1)-(5). In [5,6,10,12], the authors investigated the problem of competition with a free boundary, and in [10, 17], it is assumed that two species with weak competition propagate along the same boundary. In [16-18], a two-type model with two different free boundaries is considered for both weak and strong competition. Similar problems can be found in [2,3,19,20, 21-22].
In addition to random diffusion of the predator and the prey, the spatial-temporal variations of the predators velocity are directed by prey gradient. Several field studies measuring characteristics of individual movement confirm the basis of the hypothesis about the dependence of acceleration on a stimulus. Understanding spatial and temporal behaviors of interacting species in ecological system is a central problem in population ecology. Various types of mathematical models were proposed to study problem of predator-prey.
Many recent work [1,23, 24, 25, 26, 28, 32] studied predator–prey systems with the Leslie-Gower scheme. We will discuss the most relevant issues to our present work.
In this paper, we consider two non-linear predator-prey models of the Leslie-Gower type. The existence and specificity of a free boundary value solution are explored.