Natalia LEVASHOVA

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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.