Geodesic
Method of monotone solutions for reaction-diffusion equations
Contemporary Mathematics. Fundamental Directions, Differential and functional differential equations, Tome 63 (2017) no. 3, pp. 437-454

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Existence 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. Various applications of this method are given. 
@article{CMFD_2017_63_3_a3,
     author = {V. Volpert and V. Vougalter},
     title = {Method of monotone solutions for reaction-diffusion equations},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {437--454},
     publisher = {mathdoc},
     volume = {63},
     number = {3},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2017_63_3_a3/}
}
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V. Volpert; V. Vougalter. Method of monotone solutions for reaction-diffusion equations. Contemporary Mathematics. Fundamental Directions, Differential and functional differential equations, Tome 63 (2017) no. 3, pp. 437-454. http://geodesic.mathdoc.fr/item/CMFD_2017_63_3_a3/