Asymptotic behavior for a nonlocal diffusion equation on the half line
We study the large time behavior of solutions to a nonlocal diffusion equation, ut = J ∗ u-u with J smooth, radially symmetric and compactly supported, posed in ℝ+ with zero Dirichlet boundary conditions. In the far-field scale, ξ1 ≤ xt-1/2 ≤ ξ2 with ξ1, ξ2 > 0, the asymptotic behavior is giv...
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2015
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| Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10780947_v35_n4_p1391_Cortazar http://hdl.handle.net/20.500.12110/paper_10780947_v35_n4_p1391_Cortazar |
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paper:paper_10780947_v35_n4_p1391_Cortazar2023-06-08T16:05:34Z Asymptotic behavior for a nonlocal diffusion equation on the half line Asymptotic behavior Matched asymptotics Nonlocal diffusion We study the large time behavior of solutions to a nonlocal diffusion equation, ut = J ∗ u-u with J smooth, radially symmetric and compactly supported, posed in ℝ+ with zero Dirichlet boundary conditions. In the far-field scale, ξ1 ≤ xt-1/2 ≤ ξ2 with ξ1, ξ2 > 0, the asymptotic behavior is given by a multiple of the dipole solution for the local heat equation, hence tu(x, t) is bounded above and below by positive constants in this region for large times. The proportionality constant is determined from a conservation law, related to the asymptotic first momentum. In compact sets, after scaling the solution by a factor t3/2, it converges to a multiple of the unique stationary solution of the problem that behaves as x at infinity. The precise proportionality factor 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(t-1). 2015 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10780947_v35_n4_p1391_Cortazar http://hdl.handle.net/20.500.12110/paper_10780947_v35_n4_p1391_Cortazar |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
| collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
Asymptotic behavior Matched asymptotics Nonlocal diffusion |
| spellingShingle |
Asymptotic behavior Matched asymptotics Nonlocal diffusion Asymptotic behavior for a nonlocal diffusion equation on the half line |
| topic_facet |
Asymptotic behavior Matched asymptotics Nonlocal diffusion |
| description |
We study the large time behavior of solutions to a nonlocal diffusion equation, ut = J ∗ u-u with J smooth, radially symmetric and compactly supported, posed in ℝ+ with zero Dirichlet boundary conditions. In the far-field scale, ξ1 ≤ xt-1/2 ≤ ξ2 with ξ1, ξ2 > 0, the asymptotic behavior is given by a multiple of the dipole solution for the local heat equation, hence tu(x, t) is bounded above and below by positive constants in this region for large times. The proportionality constant is determined from a conservation law, related to the asymptotic first momentum. In compact sets, after scaling the solution by a factor t3/2, it converges to a multiple of the unique stationary solution of the problem that behaves as x at infinity. The precise proportionality factor 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(t-1). |
| title |
Asymptotic behavior for a nonlocal diffusion equation on the half line |
| title_short |
Asymptotic behavior for a nonlocal diffusion equation on the half line |
| title_full |
Asymptotic behavior for a nonlocal diffusion equation on the half line |
| title_fullStr |
Asymptotic behavior for a nonlocal diffusion equation on the half line |
| title_full_unstemmed |
Asymptotic behavior for a nonlocal diffusion equation on the half line |
| title_sort |
asymptotic behavior for a nonlocal diffusion equation on the half line |
| publishDate |
2015 |
| url |
https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10780947_v35_n4_p1391_Cortazar http://hdl.handle.net/20.500.12110/paper_10780947_v35_n4_p1391_Cortazar |
| _version_ |
1768543433439313920 |