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Neumann boundary conditions for a nonlocal nonlinear diffusion operator. Continuous and discrete models

Authors: Mauricio Bogoya, Raul Ferreira and Julio D. Rossi
Journal: Proc. Amer. Math. Soc. 135 (2007), 3837-3846
MSC (2000): Primary 35K57, 35B40
Published electronically: August 29, 2007
MathSciNet review: 2341934
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Abstract: Let $ J: \mathbb{R} \to \mathbb{R}$ be a nonnegative, smooth function with $ \int_{\mathbb{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

$\displaystyle \displaystyle u_t (x,t)= \int_{-L}^{L} \left(J\left(\dfrac{x-y}{u... ...\right) - J\left(\dfrac{x-y}{u(x,t)}\right) \right) \, dy, \quad x \in [-L,L]. $

We prove existence and uniqueness of solutions and a comparison principle. We find the asymptotic behaviour of the solutions as $ t\to \infty$: 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.

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Additional Information

Mauricio Bogoya
Affiliation: Depto. de Matemática, Univ. Católica de Chile, Santiago, Chile
Address at time of publication: Depto. de Matemática, Univ. Nacional de Colombia, Bogotá, Colombia

Raul Ferreira
Affiliation: Depto. de Matemática, U. Carlos III, 28911, Leganés, España

Julio D. Rossi
Affiliation: IMAFF, CSIC, Serrano 117, Madrid, España
Address at time of publication: Depto. Matematica, FCEyN, UBA, Buenos Aires, Argentina

Keywords: Nonlocal diffusion, Neumann boundary conditions.
Received by editor(s): April 24, 2006
Published electronically: August 29, 2007
Additional Notes: The third author was supported by University de Buenos Aires under grant TX048, by ANPCyT PICT No. 00137 and by CONICET (Argentina), MB by MECESUP (Chile) and RF by BFM2002-04572 (Spain)
Communicated by: David Tartakoff
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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