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 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 \((2 M/N ) \delta _{x=0}\) 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.

我们在度量连接的有限图上研究局部（热方程）和非局部（带可积核的卷积型问题）演化问题，其中一些边具有无限长。我们表明，对于局部和非局部问题的解的渐近行为由热方程的解给出，但是在一个星形图中，其中只有一个节点，并且无限长的边缘与原始图中一样。这样，我们在观察解的渐近行为时，可以将包含所有顶点和有限长度的所有边的紧凑分量减少到单个点。对于这个星形极限问题，解的渐近行为仅由热方程的解给出，该热方程的半线为Neumann边界条件，为\（x = 0 \）和初始基准\（（2 M / N）\ delta _ {x = 0} \）其中M是我们原始问题的初始条件的总质量，N是边界的边数无限长。此外，我们表明，当我们重新缩放内核时，非局部问题的解收敛到热方程（局部问题）的解，即找到了松弛极限。