Existence and Uniqueness of Classical Solutions to a Nonlinear Reaction-Diffusion Model
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Existence and Uniqueness of Classical Solutions to a Nonlinear Reaction-Diffusion Model A. Ambrazeviˇcius1 · V. Skakauskas1
Received: 30 April 2019 / Accepted: 16 January 2020 © Springer Nature B.V. 2020
Abstract The purpose of this paper is to investigate the existence, uniqueness, and longtime behaviour of classical solutions to a system composed of four quasilinear parabolic and three nonlinear ordinary differential equations arising in modelling of surface chemical reactions. Parabolic differential equations are solved in a domain, while the ordinary differential equations are considered on the boundary. In particular, such systems describe the enzyme-catalyzed glucose oxidation with dissolved oxygen proceeding over catalytic (enzyme-modified) surfaces. Keywords Coupled systems of parabolic and ordinary differential equation · Positive solutions · Reactions on surfaces
1 Introduction and Formulation of the Problem Coupled systems of nonlinear parabolic and ordinary differential equations determined in the same domain have been extensively studied in literature (see, [1–7, 17, 18] and references therein). In recent years attention has been given to coupled systems of nonlinear parabolic differential equations where some of them are solved in the interior of the domain while the other equations are considered on the boundary (see [8–10] and references therein). Such systems arise in modelling of surface reactions that involve the bulk diffusion of reactants toward and reaction products from the catalyst surface and surface diffusion of intermediate reaction products. In case where surface diffusion of the intermediate products is neglected, parabolic equations determined on the boundary become ordinary differential equations. In this case the coupled system is composed of parabolic equations that are solved in the interior of the domain and ordinary differential equations determined on the boundary.
B A. Ambrazeviˇcius
[email protected] V. Skakauskas [email protected]
1
Faculty of Mathematics and Informatics, Vilnius University, Naugarduko str. 24, 03225 Vilnius, Lithuania
A. Ambrazeviˇcius, V. Skakauskas
In [11], we studied a model for surface reactions between monomers and dimers, proved the existence and uniqueness theorem of its classical solution and considered its long-time behaviour. The model involves the bulk diffusion of both reactants and surface diffusion of adsorbed molecules. It is described by a coupled system of parabolic equations, where two of them a determined in the interior of the domain, while the other two are formulated on the boundary. To prove the existence of classical solution, a lower and upper solutions technique was used. In the present paper we consider a mathematical model for surface reaction which describes the glucose oxidation with dissolved oxygen proceeding over a surface modified by immobilized enzyme (glucose oxidase). From the practical point of view this reaction is directly related to the scanning electrochemical microscopy [12]. Accor
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