Asymptotic behavior for nonlocal diffusion equations

We study the asymptotic behavior for nonlocal diffusion models of the form ut = J * u - u in the whole RN or in a bounded smooth domain with Dirichlet or Neumann boundary conditions. In RN we obtain that the long time behavior of the solutions is determined by the behavior of the Fourier transform o...

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Autor principal: Rossi, Julio Daniel
Publicado: 2006
Materias:
Dirichlet boundary conditions
Fractional Laplacian
Neumann boundary conditions
Nonlocal diffusion
Acceso en línea:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00217824_v86_n3_p271_Chasseigne
http://hdl.handle.net/20.500.12110/paper_00217824_v86_n3_p271_Chasseigne
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
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spelling paper:paper_00217824_v86_n3_p271_Chasseigne2023-06-08T14:42:05Z Asymptotic behavior for nonlocal diffusion equations Rossi, Julio Daniel Dirichlet boundary conditions Fractional Laplacian Neumann boundary conditions Nonlocal diffusion We study the asymptotic behavior for nonlocal diffusion models of the form ut = J * u - u in the whole RN or in a bounded smooth domain with Dirichlet or Neumann boundary conditions. In RN we obtain that the long time behavior of the solutions is determined by the behavior of the Fourier transform of J near the origin, which is linked to the behavior of J at infinity. If over(J, ̂) (ξ) = 1 - A | ξ |α + o (| ξ |α) (0 < α ≤ 2), the asymptotic behavior is the same as the one for solutions of the evolution given by the α / 2 fractional power of the Laplacian. In particular when the nonlocal diffusion is given by a compactly supported kernel the asymptotic behavior is the same as the one for the heat equation, which is yet a local model. Concerning the Dirichlet problem for the nonlocal model we prove that the asymptotic behavior is given by an exponential decay to zero at a rate given by the first eigenvalue of an associated eigenvalue problem with profile an eigenfunction of the first eigenvalue. Finally, we analyze the Neumann problem and find an exponential convergence to the mean value of the initial condition. © 2006 Elsevier SAS. All rights reserved. Fil:Rossi, J.D. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2006 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00217824_v86_n3_p271_Chasseigne http://hdl.handle.net/20.500.12110/paper_00217824_v86_n3_p271_Chasseigne
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic Dirichlet boundary conditions
Fractional Laplacian
Neumann boundary conditions
Nonlocal diffusion
spellingShingle Dirichlet boundary conditions
Fractional Laplacian
Neumann boundary conditions
Nonlocal diffusion
Rossi, Julio Daniel
Asymptotic behavior for nonlocal diffusion equations
topic_facet Dirichlet boundary conditions
Fractional Laplacian
Neumann boundary conditions
Nonlocal diffusion
description We study the asymptotic behavior for nonlocal diffusion models of the form ut = J * u - u in the whole RN or in a bounded smooth domain with Dirichlet or Neumann boundary conditions. In RN we obtain that the long time behavior of the solutions is determined by the behavior of the Fourier transform of J near the origin, which is linked to the behavior of J at infinity. If over(J, ̂) (ξ) = 1 - A | ξ |α + o (| ξ |α) (0 < α ≤ 2), the asymptotic behavior is the same as the one for solutions of the evolution given by the α / 2 fractional power of the Laplacian. In particular when the nonlocal diffusion is given by a compactly supported kernel the asymptotic behavior is the same as the one for the heat equation, which is yet a local model. Concerning the Dirichlet problem for the nonlocal model we prove that the asymptotic behavior is given by an exponential decay to zero at a rate given by the first eigenvalue of an associated eigenvalue problem with profile an eigenfunction of the first eigenvalue. Finally, we analyze the Neumann problem and find an exponential convergence to the mean value of the initial condition. © 2006 Elsevier SAS. All rights reserved.
author Rossi, Julio Daniel
author_facet Rossi, Julio Daniel
author_sort Rossi, Julio Daniel
title Asymptotic behavior for nonlocal diffusion equations
title_short Asymptotic behavior for nonlocal diffusion equations
title_full Asymptotic behavior for nonlocal diffusion equations
title_fullStr Asymptotic behavior for nonlocal diffusion equations
title_full_unstemmed Asymptotic behavior for nonlocal diffusion equations
title_sort asymptotic behavior for nonlocal diffusion equations
publishDate 2006
url https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00217824_v86_n3_p271_Chasseigne
http://hdl.handle.net/20.500.12110/paper_00217824_v86_n3_p271_Chasseigne
work_keys_str_mv AT rossijuliodaniel asymptoticbehaviorfornonlocaldiffusionequations
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