Asymptotic behaviour for local and nonlocal evolution equations on metric graphs with some edges of infinite length
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Título: | Asymptotic behaviour for local and nonlocal evolution equations on metric graphs with some edges of infinite length |
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Autor/es: | Ignat, Liviu I. | Rossi, Julio D. | San Antolín Gil, Ángel |
Centro, Departamento o Servicio: | Universidad de Alicante. Departamento de Matemáticas |
Palabras clave: | Nonlocal diffusion | Local diffusion | Quantum graphs | Compactness arguments | Asymptotic behaviour |
Área/s de conocimiento: | Análisis Matemático |
Fecha de publicación: | jun-2021 |
Editor: | Springer Nature |
Cita bibliográfica: | Annali di Matematica Pura ed Applicata (1923 -). 2021, 200: 1301-1339. https://doi.org/10.1007/s10231-020-01039-5 |
Resumen: | We study local (the heat equation) and nonlocal (convolution-type problems with an integrable kernel) evolution problems on a metric connected finite graph in which some of the edges have infinity length. We show that the asymptotic behaviour of the solutions to both local and nonlocal problems is given by the solution of the heat equation, but on a starshaped graph in which there are only one node and as many infinite edges as in the original graph. In this way, we obtain that the compact component that consists in all the vertices and all the edges of finite length can be reduced to a single point when looking at the asymptotic behaviour of the solutions. For this star-shaped limit problem, the asymptotic behaviour of the solutions is just given by the solution to the heat equation in a half line with a Neumann boundary condition at x = 0 and initial datum (2M/N)δx=0 where M is the total mass of the initial condition for our original problem and N is the number of edges of infinite length. In addition, we show that solutions to the nonlocal problem converge, when we rescale the kernel, to solutions to the heat equation (the local problem), that is, we find a relaxation limit. |
Patrocinador/es: | L. I. was partially supported by a grant of Ministry of Research and Innovation, CNCSUEFISCDI, project PN-III-P1-1.1-TE-2016- 2233, within PNCDI III. J.D.R. partially supported by CONICET grant PIP GI No 11220150100036CO (Argentina), PICT-2018-03183 (Argentina) and UBACyT grant 20020160100155BA (Argentina). |
URI: | http://hdl.handle.net/10045/114106 |
ISSN: | 0373-3114 (Print) | 1618-1891 (Online) |
DOI: | 10.1007/s10231-020-01039-5 |
Idioma: | eng |
Tipo: | info:eu-repo/semantics/article |
Derechos: | © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020 |
Revisión científica: | si |
Versión del editor: | https://doi.org/10.1007/s10231-020-01039-5 |
Aparece en las colecciones: | Personal Investigador sin Adscripción a Grupo INV - GAM - Artículos de Revistas |
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Ignat_etal_2021_AnnalidiMatematica_final.pdf | Versión final (acceso restringido) | 4,17 MB | Adobe PDF | Abrir Solicitar una copia |
Ignat_etal_2021_AnnalidiMatematica_preprint.pdf | Preprint (acceso abierto) | 376,98 kB | Adobe PDF | Abrir Vista previa |
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