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Abstract:

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.

Registro:

Documento: Artículo
Título:Asymptotic behavior for nonlocal diffusion equations
Autor:Chasseigne, E.; Chaves, M.; Rossi, J.D.
Filiación:Laboratoire de Mathématiques et Physique Théorique, Université de Tours, Parc de Grandmont, F-37200 Tours, France
Depto. Matemática Aplicada, Universidad de Salamanca, Salamanca, Spain
Instituto de Matemáticas y Física Fundamental, Consejo Superior de Investigaciones Científicas, Serrano 123, Madrid, Spain
Depto. Matemática, FCEyN, Universidad de Buenos Aires, Argentina
Palabras clave:Dirichlet boundary conditions; Fractional Laplacian; Neumann boundary conditions; Nonlocal diffusion
Año:2006
Volumen:86
Número:3
Página de inicio:271
Página de fin:291
DOI: http://dx.doi.org/10.1016/j.matpur.2006.04.005
Título revista:Journal des Mathematiques Pures et Appliquees
Título revista abreviado:J. Math. Pures Appl.
ISSN:00217824
CODEN:JMPAA
PDF:https://bibliotecadigital.exactas.uba.ar/download/paper/paper_00217824_v86_n3_p271_Chasseigne.pdf
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00217824_v86_n3_p271_Chasseigne

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Citas:

---------- APA ----------
Chasseigne, E., Chaves, M. & Rossi, J.D. (2006) . Asymptotic behavior for nonlocal diffusion equations. Journal des Mathematiques Pures et Appliquees, 86(3), 271-291.
http://dx.doi.org/10.1016/j.matpur.2006.04.005
---------- CHICAGO ----------
Chasseigne, E., Chaves, M., Rossi, J.D. "Asymptotic behavior for nonlocal diffusion equations" . Journal des Mathematiques Pures et Appliquees 86, no. 3 (2006) : 271-291.
http://dx.doi.org/10.1016/j.matpur.2006.04.005
---------- MLA ----------
Chasseigne, E., Chaves, M., Rossi, J.D. "Asymptotic behavior for nonlocal diffusion equations" . Journal des Mathematiques Pures et Appliquees, vol. 86, no. 3, 2006, pp. 271-291.
http://dx.doi.org/10.1016/j.matpur.2006.04.005
---------- VANCOUVER ----------
Chasseigne, E., Chaves, M., Rossi, J.D. Asymptotic behavior for nonlocal diffusion equations. J. Math. Pures Appl. 2006;86(3):271-291.
http://dx.doi.org/10.1016/j.matpur.2006.04.005