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We study local (the heat equation) and nonlocal (convolution type problemswith an integrable kernel) evolution problems on a metric connected finitegraph in which some of the edges have infinity length. We show that theasymptotic behaviour of the solutions to both local and nonlocal problems isgiven by the solution of the heat equation, but on a star shaped graph in whichthere is only one node and as many infinite edges as in the original graph. Inthis way we obtain that the compact component that consists in all the verticesand all the edges of finite length can be reduced to a single point whenlooking at the asymptotic behaviour of the solutions. For this star shapedlimit problem the asymptotic behaviour of the solutions is just given by thesolution to the heat equation in a half line with a Neumann boundary conditionat $x = 0$ and initial datum $(2M/N)\delta_{x=0}$ where M is the total mass ofthe initial condition for our original problem and $N$ is the number of edgesof infinite length. In addition, we show that solutions to the nonlocal problemconverge, when we rescale the kernel, to solutions to the heat equation (thelocal problem), that is, we find a relaxation limit.

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