Delay nonlinear boundary conditions as limit of reactions concentrating in the boundary |
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Authors: | Gleiciane da Silva Aragão Sergio Muniz Oliva |
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Institution: | 1. Departamento de Ciências Exatas e da Terra, Universidade Federal de São Paulo, Rua Prof. Artur Riedel, 275, Jardim Eldorado, CEP 09972-270, Diadema SP, Brazil;2. Departamento de Matemática Aplicada, Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1010, Cidade Universitária, CEP 05508-090, São Paulo SP, Brazil |
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Abstract: | In this work, we are interested in the dynamic behavior of a parabolic problem with nonlinear boundary conditions and delay in the boundary. We construct a reaction–diffusion problem with delay in the interior, where the reaction term is concentrated in a neighborhood of the boundary and this neighborhood shrinks to boundary, as a parameter ? goes to zero. We analyze the limit of the solutions of this concentrated problem and prove that these solutions converge in certain continuous function spaces to the unique solution of the parabolic problem with delay in the boundary. This convergence result allows us to approximate the solution of equations with delay acting on the boundary by solutions of equations with delay acting in the interior and it may contribute to analyze the dynamic behavior of delay equations when the delay is at the boundary. |
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