Contrasting structures as solutions of the reaction-diffusion problem with an arbitrary roots order

In this paper we consider the initial -- boundary value problem, based on the reaction-diffusion equation in a homogeneous and inhomogeneous medium This equation possess the solutions known as a non-stationary contrasting structures. The right side of the equation is a given smooth function with three ordered roots. Moreover, in the vicinity of the first and the last roots this function tends to zero faster than the first power of the coordinate. It can be also be a power function with the exponent greater than one. It can also be a function that decreases faster than any power of the coordinate, for example, exponential. Thus this paper generalises the results of a series of papers by V. F. Butuzov and his colleagues. Also we discuss the new problem in which the density function is power -- exponential. We show that the nose and tail regions of the contrasting structure tend to their stationary levels in different ways. The tail part tends to the stationary level more slowly than the nose region. The results are strictly substantiated by the differential inequalities method. The use of differential inequalities for the reaction -- diffusion problems with a small parameter was developed H. N. Nefedov. The results are consistent with the results of computer simulation also presented in the paper.