Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Diffusion systems reacting at the boundary


Author: W. E. Olmstead
Journal: Quart. Appl. Math. 38 (1980), 51-59
MSC: Primary 80A30; Secondary 35K50
DOI: https://doi.org/10.1090/qam/575832
MathSciNet review: 575832
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Abstract: The problem considered is that of two species or chemical concentrations which independently diffuse within the same or adjacent regions. The coupling interaction takes place only along a common boundary. This boundary reaction is allowed to be either totally dissipative wherein both species are removed by the interaction, or semi-dissipative wherein one species is stimulated at the expense of the other. This physical situation is modeled by two independent, linear heat equations, each defined over a one-dimensional, semi-infinite domain. Associated with each heat equation is a boundary flux condition containing a nonlinear interactive term which couples the solutions of the two heat equations. With only boundary interaction, the problem can be reduced to the study of two coupled Volterra integral equations. By using monotone operator methods these integral equations are shown to have positive solutions. Uniqueness is also established. The large-time asymptotic behavior of the solutions is examined for the cases of both fast and slow decay of data.


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DOI: https://doi.org/10.1090/qam/575832
Article copyright: © Copyright 1980 American Mathematical Society


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