ИСТИНА

A system of two diffusion equations is considered, which underlies the autowave model of the megalopolises development. In this model, we consider the urban density as the autowave front. On the border of urbanized areas, the solution to the problem has a large gradient, rapidly changing from values ​​corresponding to urban development to almost zero in rural areas. A scheme for constructing an asymptotic approximation of a solution with a large gradient and a method for constructing the upper and lower solutions as modifications of the asymptotic approximation are presented. To obtain conditions for the existence of solutions with a large gradient for the system of equations under consideration, as well as conditions for the stability of the corresponding stationary system solutions, the asymptotic method of differential inequalities is used, in the course of which the constructed upper and lower solutions are used. The obtained conditions of existence and stability should be used while creating autowave models.