Abstract:
Let J: ℝ → ℝ be a nonnegative, smooth function with ∫ℝ J(r)dr = 1, supported in [-1, 1], symmetric, J(r) = J(-r), and strictly increasing in [-1,0]. We consider the Neumann boundary value problem for a nonlocal, nonlinear operator that is similar to the porous medium, and we study the equation ut(x, t)=∫L-L(J(x-y/ u(y,t) - J(x-y/u(x, t))dy, x∈[-L, L].We prove existence and uniqueness of solutions and a comparison principle. We find the asymptotic behaviour of the solutions as t → ∞: they converge to the mean value of the initial data. Next, we consider a discrete version of the above problem. Under suitable hypotheses we prove that the discrete model has properties analogous to the continuous one. Moreover, solutions of the discrete problem converge to the continuous ones when the mesh parameter goes to zero. Finally, we perform some numerical experiments. © 2007 American Mathematical Society.
Registro:
Documento: |
Artículo
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Título: | Neumann boundary conditions for a nonlocal nonlinear diffusion operator. Continuous and discrete models |
Autor: | Bogoya, M.; Ferreira, R.; Rossi, J.D. |
Filiación: | Depto. de Matemática, Univ. Católica de Chile, Santiago, Chile Depto. de Matemática, Univ. Nacional de Colombia, Bogotá, Colombia Depto. de Matemática, U. Carlos III, 28911, Leganés, Spain IMAFF, CSIC, Serrano 117, Madrid, Spain Depto. Matematica, FCEyN, UBA, Buenos Aires, Argentina
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Palabras clave: | Neumann boundary conditions; Nonlocal diffusion |
Año: | 2007
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Volumen: | 135
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Número: | 12
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Página de inicio: | 3837
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Página de fin: | 3846
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DOI: |
http://dx.doi.org/10.1090/S0002-9939-07-09205-2 |
Título revista: | Proceedings of the American Mathematical Society
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Título revista abreviado: | Proc. Am. Math. Soc.
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ISSN: | 00029939
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PDF: | https://bibliotecadigital.exactas.uba.ar/download/paper/paper_00029939_v135_n12_p3837_Bogoya.pdf |
Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00029939_v135_n12_p3837_Bogoya |
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Citas:
---------- APA ----------
Bogoya, M., Ferreira, R. & Rossi, J.D.
(2007)
. Neumann boundary conditions for a nonlocal nonlinear diffusion operator. Continuous and discrete models. Proceedings of the American Mathematical Society, 135(12), 3837-3846.
http://dx.doi.org/10.1090/S0002-9939-07-09205-2---------- CHICAGO ----------
Bogoya, M., Ferreira, R., Rossi, J.D.
"Neumann boundary conditions for a nonlocal nonlinear diffusion operator. Continuous and discrete models"
. Proceedings of the American Mathematical Society 135, no. 12
(2007) : 3837-3846.
http://dx.doi.org/10.1090/S0002-9939-07-09205-2---------- MLA ----------
Bogoya, M., Ferreira, R., Rossi, J.D.
"Neumann boundary conditions for a nonlocal nonlinear diffusion operator. Continuous and discrete models"
. Proceedings of the American Mathematical Society, vol. 135, no. 12, 2007, pp. 3837-3846.
http://dx.doi.org/10.1090/S0002-9939-07-09205-2---------- VANCOUVER ----------
Bogoya, M., Ferreira, R., Rossi, J.D. Neumann boundary conditions for a nonlocal nonlinear diffusion operator. Continuous and discrete models. Proc. Am. Math. Soc. 2007;135(12):3837-3846.
http://dx.doi.org/10.1090/S0002-9939-07-09205-2