INTRODUCTION Propagating fronts are characteristic for many physical phenomena. In case of reaction-diffusion-advection processes, these can be combustion or strain fronts. Solutions having large gradients to problems of this type also arise in nonlinear acoustics. Such problems include, for example, the Burgers equation, as well as equations with modular nonlinearity. The stationary reaction-diffusion-advection equations can be used for modelling of wind field distribution in the presence of plant heterogeneity. The domain where the solution has a large gradient is called the internal transition layer. The numerical implementation of solutions to problems with internal transition layers requires a preliminary analysis of the existence conditions and stability. In particular, for the numerical solution of some applied problems, the calculation method for establishing is often used, when the solution of the boundary value problem for the elliptic equation is found numerically as the solution of the corresponding initial-boundary value problem for the parabolic equation over a sufficiently long period of time. To implement this method, information on the asymptotic stability and the domain of attraction of the stationary solution is needed. In this paper, we consider the initial-boundary-value problem for reaction-diffusion-advection equation and the question of its moving front type solution stabilizing over an infinitely large time interval to the solution of the corresponding stationary problem. The existence of moving front solution is investigated in. The existence conditions of an asymptotically stable solution to the stationary problem are known from. To prove the stabilization theorem, in this paper we use the method of upper and lower solutions, which for this class of problems is justified in. The main idea of the proof is to show that the upper and lower solutions of the initial-boundary-value problem on an asymptotically large time interval fall into the attraction domain of the stationary solution. The upper and lower solutions with large gradients in the region of the internal transition layer are constructed according to the asymptotic method of differential inequalities as modifications of asymptotic approximations of the solutions to these problems in a small parameter. A small parameter here is the width of the inner transition layer with respect to the width of the front propagation region. The study conducted in this work gives an answer about non-local domain of attraction of the stationary solution. In addition, an estimate of the time interval is obtained in which the solution of the front type falls into the local domain of attraction of the stationary solution, that is, in fact, the criterion for the numerical solution stationing.
INTRODUCTION The article considers a system of two second order nonlinear differential equations with discontinuous functions on the right sides. The aim of the work is to obtain sufficient conditions for the existence, local uniqueness, and asymptotic stability of a stationary solution of a parabolic system with a large gradient in the vicinity of the discontinuity points of the right-hand sides. The area where the function undergoes large gradient is called the internal transition layer. The authors arrived at this formulation of the problem during the development of the autowave model for the development of megacities . This model is based on the activator-inhibitor system of two equations where the urban area acts as the activator, and the inhibitor is determined by environmental or economic factors due to urban planning policies of a country. The presence of barriers that prevent the propagation of the front of the activator, for example, large bodies of water, is taken into account in the model as a jump in the functions on the right-hand sides. Obviously, the numerical solution of such a problem should be preceded by an analytical study of the existence of the mentioned solution, which was done in the present work. The proof of the existence and asymptotic stability of the stationary solution of the initial-boundary-value problem here is carried out using the asymptotic method of differential inequalities , based on the method of super- and subsolutions.The latter was extended to problems with a single discontinuity point of the first kind on the right-hand sides of the equations based on a modified proof of the corresponding theorem from , where it was carried out for the case of C² continuous right-hand sides.