publication . Article . Preprint . 2012

Asymptotic behavior for a nonlocal diffusion equation in domains with holes

Carmen Cortázar; Manuel Elgueta; Fernando Quirós; Noemi Wolanski;
Open Access English
  • Published: 01 Jan 2012
  • Publisher: SPRINGER
Abstract
The paper deals with the asymptotic behavior of solutions to a non-local diffusion equation, ut = J ∗u −u := Lu, in an exterior domain, Ω, which excludes one or several holes, and with zero Dirichlet data on RN \Ω. When the space dimension is three or more this behavior is given by a multiple of the fundamental solution of the heat equation away from the holes. On the other hand, if the solution is scaled according to its decay factor, close to the holes it behaves like a function that is L-harmonic, Lu = 0, in the exterior domain and vanishes in its complement. The height of such a function at infinity is determined through a matching procedure with the multipl...
Subjects
free text keywords: Mathematics - Analysis of PDEs, 35R09, 45K05, 45M05, Nonlocal diffusion, Exterior domain, Asymptotic behavior, Matched asymptotics, Matemática Pura, Matemáticas, CIENCIAS NATURALES Y EXACTAS, Mechanical Engineering, Mathematics (miscellaneous), Analysis, Mathematical analysis, Omega, Infinity, media_common.quotation_subject, media_common, Complex system, Dirichlet distribution, symbols.namesake, symbols, Heat equation, Mathematics, Fundamental solution, Diffusion equation, Asymptotic analysis
16 references, page 1 of 2

[1] Andreu, F.; Mazo´n, J.M.; Rossi, J.D.; Toledo, J., “Nonlocal Diffusion Problems”. American Mathematical Society. Mathematical Surveys and Monographs 2010. Vol. 165. ISBN-10: 0-8218-5230-2. ISBN-13: 978-0-8218-5230-9.

[2] Bates, P.W.; Chmaj, A. An integrodifferential model for phase transitions: stationary solutions in higher dimensions. J. Statist. Phys. 95 (1999), no. 5-6, 1119-1139. [OpenAIRE]

[3] Bates, P.W.; Chmaj, A. A discrete convolution model for phase transitions, Arch. Ration. Mech. Anal. 150 (1999), no. 4, 281-305. [OpenAIRE]

[4] Bates, P.W.; Zhao, G. Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal. J. Math. Anal. Appl. 332 (2007), no. 1, 428-440.

[5] Br¨andle, C.; Chasseigne, E.; Ferreira, R. Unbounded solutions of the nonlocal heat equation. Commun. Pure Appl. Anal. 10 (2011), no. 6, 1663-1686.

[6] Br¨andle, C.; Quir´os, F.; V´azquez, J. L. Asymptotic behaviour of the porous media equation in domains with holes. Interfaces Free Bound. 9 (2007), no. 2, 211-232.

[7] Carrillo, C.; Fife, P. Spatial effects in discrete generation population models. J. Math. Biol. 50 (2005), no. 2, 161-188.

[8] Chasseigne, E.; Chaves, M.; Rossi, J.D. Asymptotic behavior for nonlocal diffusion equations. J. Math. Pures Appl. (9) 86 (2006), no. 3, 271-291.

[9] Cort´azar, C.; Elgueta, M.; Rossi, J.D.; Wolanski, N. How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems. Arch. Ration. Mech. Anal. 187 (2008), no. 1, 137-156.

[10] Gilboa, G.; Osher, S. Nonlocal operators with application to image processing. Multiscale Model. Simul. 7 (2008), no. 3, 1005-1028.

[11] Herraiz, L.A. A nonlinear parabolic problem in an exterior domain. J. Differential Equations 142 (1998), no. 2, 371-412. [OpenAIRE]

[12] Iagar, R. G.; V´azquez, J.L. Asymptotic analysis of the p-Laplacian flow in an exterior domain. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 26 (2009), no. 2, 497-520.

[13] Ignat, L.I.; Rossi, J.D. Refined asymptotic expansions for nonlocal diffusion equations. J. Evol. Equ. 8 (2008), no. 4, 617-629.

[14] Fife, P. Some nonclassical trends in parabolic and parabolic-like evolutions, in “Trends in nonlinear analysis”, pp. 153-191, Springer-Verlag, Berlin, 2003. [OpenAIRE]

[15] Lederman, C.; Wolanski, N. Singular perturbation in a nonlocal diffusion model, Communications in PDE 31 (2006), no. 2 195-241.

16 references, page 1 of 2
Abstract
The paper deals with the asymptotic behavior of solutions to a non-local diffusion equation, ut = J ∗u −u := Lu, in an exterior domain, Ω, which excludes one or several holes, and with zero Dirichlet data on RN \Ω. When the space dimension is three or more this behavior is given by a multiple of the fundamental solution of the heat equation away from the holes. On the other hand, if the solution is scaled according to its decay factor, close to the holes it behaves like a function that is L-harmonic, Lu = 0, in the exterior domain and vanishes in its complement. The height of such a function at infinity is determined through a matching procedure with the multipl...
Subjects
free text keywords: Mathematics - Analysis of PDEs, 35R09, 45K05, 45M05, Nonlocal diffusion, Exterior domain, Asymptotic behavior, Matched asymptotics, Matemática Pura, Matemáticas, CIENCIAS NATURALES Y EXACTAS, Mechanical Engineering, Mathematics (miscellaneous), Analysis, Mathematical analysis, Omega, Infinity, media_common.quotation_subject, media_common, Complex system, Dirichlet distribution, symbols.namesake, symbols, Heat equation, Mathematics, Fundamental solution, Diffusion equation, Asymptotic analysis
16 references, page 1 of 2

[1] Andreu, F.; Mazo´n, J.M.; Rossi, J.D.; Toledo, J., “Nonlocal Diffusion Problems”. American Mathematical Society. Mathematical Surveys and Monographs 2010. Vol. 165. ISBN-10: 0-8218-5230-2. ISBN-13: 978-0-8218-5230-9.

[2] Bates, P.W.; Chmaj, A. An integrodifferential model for phase transitions: stationary solutions in higher dimensions. J. Statist. Phys. 95 (1999), no. 5-6, 1119-1139. [OpenAIRE]

[3] Bates, P.W.; Chmaj, A. A discrete convolution model for phase transitions, Arch. Ration. Mech. Anal. 150 (1999), no. 4, 281-305. [OpenAIRE]

[4] Bates, P.W.; Zhao, G. Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal. J. Math. Anal. Appl. 332 (2007), no. 1, 428-440.

[5] Br¨andle, C.; Chasseigne, E.; Ferreira, R. Unbounded solutions of the nonlocal heat equation. Commun. Pure Appl. Anal. 10 (2011), no. 6, 1663-1686.

[6] Br¨andle, C.; Quir´os, F.; V´azquez, J. L. Asymptotic behaviour of the porous media equation in domains with holes. Interfaces Free Bound. 9 (2007), no. 2, 211-232.

[7] Carrillo, C.; Fife, P. Spatial effects in discrete generation population models. J. Math. Biol. 50 (2005), no. 2, 161-188.

[8] Chasseigne, E.; Chaves, M.; Rossi, J.D. Asymptotic behavior for nonlocal diffusion equations. J. Math. Pures Appl. (9) 86 (2006), no. 3, 271-291.

[9] Cort´azar, C.; Elgueta, M.; Rossi, J.D.; Wolanski, N. How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems. Arch. Ration. Mech. Anal. 187 (2008), no. 1, 137-156.

[10] Gilboa, G.; Osher, S. Nonlocal operators with application to image processing. Multiscale Model. Simul. 7 (2008), no. 3, 1005-1028.

[11] Herraiz, L.A. A nonlinear parabolic problem in an exterior domain. J. Differential Equations 142 (1998), no. 2, 371-412. [OpenAIRE]

[12] Iagar, R. G.; V´azquez, J.L. Asymptotic analysis of the p-Laplacian flow in an exterior domain. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 26 (2009), no. 2, 497-520.

[13] Ignat, L.I.; Rossi, J.D. Refined asymptotic expansions for nonlocal diffusion equations. J. Evol. Equ. 8 (2008), no. 4, 617-629.

[14] Fife, P. Some nonclassical trends in parabolic and parabolic-like evolutions, in “Trends in nonlinear analysis”, pp. 153-191, Springer-Verlag, Berlin, 2003. [OpenAIRE]

[15] Lederman, C.; Wolanski, N. Singular perturbation in a nonlocal diffusion model, Communications in PDE 31 (2006), no. 2 195-241.

16 references, page 1 of 2
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