Contemporary Mathematics. Fundamental DirectionsContemporary Mathematics. Fundamental DirectionsСовременная математика. Фундаментальные направления2413-36392949-0618Peoples’ Friendship University of Russia named after Patrice Lumumba (RUDN University)2239210.22363/2413-3639-2017-63-3-437-454New ResultsНовые результатыResearch ArticleMethod of Monotone Solutions for Reaction-Diffusion EquationsМетод монотонных решений для уравнений реакции-диффузииVolpertVВольпертВvolpert@math.univ-lyon1.frVougalterVВугальтерВvolpert@math.univ-lyon1.frInstitut Camille Jordan, UMR 5208 CNRS, University LyonINRIA Team Dracula, INRIA Lyon La DouaRUDN UniversityРоссийский университет дружбы народовUniversity of Toronto15122017633Differential and Functional Differential EquationsДифференциальные и функционально-дифференциальные уравнения43745406122019Copyright ©; 2019, Contemporary Mathematics. Fundamental DirectionsCopyright ©; 2019, Современная математика. Фундаментальные направления2019Contemporary Mathematics. Fundamental DirectionsСовременная математика. Фундаментальные направленияhttps://journals.rudn.ru/CMFD/article/view/22392Existence of solutions of reaction-diffusion systems of equations in unbounded domains is studied by the Leray-Schauder (LS) method based on the topological degree for elliptic operators in unbounded domains and on a priori estimates of solutions in weighted spaces. We identify some reactiondiffusion systems for which there exist two subclasses of solutions separated in the function space, monotone and non-monotone solutions. A priori estimates and existence of solutions are obtained for monotone solutions allowing to prove their existence by the LS method. 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