Asymptotic behaviour for local and nonlocal evolution equations on metric graphs with some edges of infinite length
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Asymptotic behaviour for local and nonlocal evolution equations on metric graphs with some edges of infinite length Liviu I. Ignat1,2 · Julio D. Rossi3 · Angel San Antolin4 Received: 25 May 2020 / Accepted: 12 September 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract 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. Keywords Nonlocal diffusion · Local diffusion · Quantum graphs · Compactness arguments · Asymptotic behaviour Mathematics Subject Classification 35B40 · 45G10 · 46B50
1 Introduction The aim of this paper is to study solutions to diffusion equations both local and nonlocal in a metric graph. A metric graph is by definition a combinatorial graph where the edges, denoted by {ej } , are considered as intervals of the real line {Ij } with a distance on each one of them.
* Julio D. Rossi [email protected] http://mate.dm.uba.ar Extended author information available on the last page of the article
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These edges/intervals are glued together according to the combinatorial structure. We assume here that at least one of the edges is not bounded. (It has infinite length.) Metric graphs have received lot of attention in recent years both from the point of view of pure mathematicians and also from potential applications. The name quantum graph is used for a graph considered as an one-dimensional singular variety and equipped with a differential operator (local or in some cases nonlocal). There are several reasons for studying quantum graphs. They naturally arise as simplified (due to reduced dimension) models in mathematics, physics, chemistry, and engineering (e.g. nanotechnology and microelectronics), when one considers propagation of waves of various natures (electromagnetic, acoustic, etc.) through a quasione-dimensional system (often a mesoscopic one) that lo
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