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Method of Monotone Solutions for Reaction-Diffusion Equations

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Abstract

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 reaction-diffusion systems for which there exist two subclasses of solutions separated in the function space, monotone and nonmonotone 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.

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Authors and Affiliations

  1. Institut Camille Jordan, University Lyon 1, Villeurbanne, France

    V. Volpert

  2. INRIA Team Dracula, INRIA Lyon La Doua, Villeurbanne, France

    V. Volpert

  3. Peoples’ Friendship University of Russia (RUDN University), Moscow, Russia

    V. Volpert

  4. University of Toronto, Toronto, Canada

    V. Vougalter

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  1. V. Volpert
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Correspondence to V. Volpert.

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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 63, No. 3, Differential and Functional Differential Equations, 2017.

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Volpert, V., Vougalter, V. Method of Monotone Solutions for Reaction-Diffusion Equations. J Math Sci 253, 660–675 (2021). https://doi.org/10.1007/s10958-021-05260-2

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