Artículo
Abstract:
We study the long time behavior of bounded, integrable solutions to a nonlocal diffusion equation, ∂tu=J*u-u, where J is a smooth, radially symmetric kernel with support Bd(0)⊂R2. The problem is set in an exterior two-dimensional domain which excludes a hole H, and with zero Dirichlet data on H. In the far field scale, ξ1≤|x|t-1/2≤ξ2 with ξ1, ξ2>0, the scaled function logtu(x, t) behaves as a multiple of the fundamental solution for the local heat equation with a certain diffusivity determined by J. The proportionality constant, which characterizes the first non-trivial term in the asymptotic behavior of the mass, is given by means of the asymptotic 'logarithmic momentum' of the solution, limt→∞∫R2u(x,t)log|x|dx. This asymptotic quantity can be easily computed in terms of the initial data. In the near field scale, |x|≤t1/2h(t) with limt→∞h(t)=0, the scaled function t(logt)2u(x, t)/log|x| converges to a multiple of φ(x)/log|x|, where φ is the unique stationary solution of the problem that behaves as log|x| when |x|→∞. The proportionality constant is obtained through a matching procedure with the far field limit. Finally, in the very far field, |x|≥t1/2g(t) with g(t)→∞, the solution is proved to be of order o((tlogt)-1). © 2015 Elsevier Inc.
Registro:
| Documento: | Artículo |
| Título: | Asymptotic behavior for a nonlocal diffusion equation in exterior domains: The critical two-dimensional case |
| Autor: | Cortázar, C.; Elgueta, M.; Quirós, F.; Wolanski, N. |
| Filiación: | Departamento de Matemática, Pontificia Universidad Católica de Chile, Santiago, Chile Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid, 28049, Spain Departamento de Matemática, FCEyN, UBA, Ciudad Universitaria, Pab. I, Buenos Aires, 1428, Argentina IMAS, CONICET, Ciudad Universitaria, Pab. I, Buenos Aires, 1428, Argentina |
| Palabras clave: | Asymptotic behavior; Exterior domain; Matched asymptotics; Nonlocal diffusion |
| Año: | 2016 |
| Volumen: | 436 |
| Número: | 1 |
| Página de inicio: | 586 |
| Página de fin: | 610 |
| DOI: | http://dx.doi.org/10.1016/j.jmaa.2015.12.021 |
| Título revista: | Journal of Mathematical Analysis and Applications |
| Título revista abreviado: | J. Math. Anal. Appl. |
| ISSN: | 0022247X |
| Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0022247X_v436_n1_p586_Cortazar |
Referencias:
- Bates, P.W., Chmaj, A., An integrodifferential model for phase transitions: stationary solutions in higher dimensions (1999) J. Stat. Phys., 95 (5-6), pp. 1119-1139
- Bates, P.W., Chmaj, A., A discrete convolution model for phase transitions (1999) Arch. Ration. Mech. Anal., 150 (4), pp. 281-305
- Bates, P.W., Zhao, G., Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal (2007) J. Math. Anal. Appl., 332 (1), pp. 428-440
- Brändle, C., Chasseigne, E., Quirós, F., Phase transitions with midrange interactions: a nonlocal Stefan model (2012) SIAM J. Math. Anal., 44 (4), pp. 3071-3100
- Carrillo, C., Fife, P., Spatial effects in discrete generation population models (2005) J. Math. Biol., 50 (2), pp. 161-188
- Chasseigne, E., Chaves, M., Rossi, J.D., Asymptotic behavior for nonlocal diffusion equations (2006) J. Math. Pures Appl. (9), 86 (3), pp. 271-291
- Cortázar, C., Elgueta, M., Quirós, F., Wolanski, N., Asymptotic behavior for a nonlocal diffusion equation in domains with holes (2012) Arch. Ration. Mech. Anal., 205 (2), pp. 673-697
- Cortázar, C., Elgueta, M., Quirós, F., Wolanski, N., Asymptotic behavior for a nonlocal diffusion equation on the half line (2015) Discrete Contin. Dyn. Syst., 35 (4), pp. 1391-1407
- Cortázar, C., Elgueta, M., Quirós, F., Wolanski, N., Asymptotic behavior for a one-dimensional nonlocal diffusion equation in exterior domains, , arxiv:1412.0731, preprint
- Fife, P., Some nonclassical trends in parabolic and parabolic-like evolutions (2003) Trends in Nonlinear Analysis, pp. 153-191. , Springer-Verlag, Berlin
- Gilboa, G., Osher, S., Nonlocal operators with application to image processing (2008) Multiscale Model. Simul., 7 (3), pp. 1005-1028
- Herraiz, L., Asymptotic behaviour of solutions of some semilinear parabolic problems (1999) Ann. Inst. H. Poincaré Anal. Non Linéaire, 16 (1), pp. 49-105
- Ignat, L.I., Rossi, J.D., Refined asymptotic expansions for nonlocal diffusion equations (2008) J. Evol. Equ., 8 (4), pp. 617-629
- Terra, J., Wolanski, N., Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data (2011) Discrete Contin. Dyn. Syst., 31 (2), pp. 581-605
Citas:
---------- APA ----------
Cortázar, C., Elgueta, M., Quirós, F. & Wolanski, N. (2016) . Asymptotic behavior for a nonlocal diffusion equation in exterior domains: The critical two-dimensional case. Journal of Mathematical Analysis and Applications, 436(1), 586-610.http://dx.doi.org/10.1016/j.jmaa.2015.12.021
---------- CHICAGO ----------
Cortázar, C., Elgueta, M., Quirós, F., Wolanski, N. "Asymptotic behavior for a nonlocal diffusion equation in exterior domains: The critical two-dimensional case" . Journal of Mathematical Analysis and Applications 436, no. 1 (2016) : 586-610.http://dx.doi.org/10.1016/j.jmaa.2015.12.021
---------- MLA ----------
Cortázar, C., Elgueta, M., Quirós, F., Wolanski, N. "Asymptotic behavior for a nonlocal diffusion equation in exterior domains: The critical two-dimensional case" . Journal of Mathematical Analysis and Applications, vol. 436, no. 1, 2016, pp. 586-610.http://dx.doi.org/10.1016/j.jmaa.2015.12.021
---------- VANCOUVER ----------
Cortázar, C., Elgueta, M., Quirós, F., Wolanski, N. Asymptotic behavior for a nonlocal diffusion equation in exterior domains: The critical two-dimensional case. J. Math. Anal. Appl. 2016;436(1):586-610.http://dx.doi.org/10.1016/j.jmaa.2015.12.021
