Asymptotic behaviour for local and nonlocal evolution equations on metric graphs with some edges of infinite length

TL;DR

This paper analyzes the long-term behavior of local and nonlocal evolution equations on metric graphs with some edges extending infinitely, showing they converge to solutions on a simplified star-shaped graph with a specific boundary condition.

Contribution

It establishes the asymptotic equivalence of solutions on complex graphs with infinite edges to solutions on a simplified star-shaped graph, including a relaxation limit for nonlocal problems.

Findings

Solutions asymptotically behave like heat equation solutions on a star graph

The finite component reduces to a single point in the limit

Nonlocal solutions converge to local heat equation solutions under kernel rescaling

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 star shaped graph in which there is 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 $(2M/N)\delta_{x=0}$ 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.