Exact Solutions of the Equations of a Nonstationary Front with Equilibrium Points of an Infinite Order of Degeneracy

Abstract

A family of exact solutions for a quasilinear evolution equation that describes the reaction–diffusion process in a medium with an infinite order of degeneracy of the two extreme roots of the source density function is found. Several terms of the formal asymptotic series are constructed for solving the type of a contrast structure that represents the solution of the initial-boundary value problem in a spatially homogeneous medium for the case of a Gaussian source density function in the neighborhood of the extreme roots. The correctness of the partial sum of the asymptotic series using the method of differential inequalities is justified. It is shown that the leading edge of the moving front of the contrast structure is exponential and the trailing edge of the front is represented by a much more slowly decreasing function, which is expressed by a power function of the logarithm of the coordinate for the Gaussian source density function.