论文标题
对于一定长度边缘的度量图上局部和非局部进化方程的渐近行为
Asymptotic behaviour for local and nonlocal evolution equations on metric graphs with some edges of infinite length
论文作者
论文摘要
我们在公制连接的有限图上研究局部（热方程）和非局部（卷积类型问题）的演变问题，其中某些边缘具有无穷大长度。我们表明，溶液对局部和非局部问题的渐近行为是由热方程式的溶液给出的，但是在一个星形的图上，只有一个节点和与原始图中的无限边缘一样多。通过这种方式，我们可以获得在所有顶点和有限长度的所有边缘组成的紧凑型组件时，可以在查看溶液的渐近行为时将其降低至单点。对于这个星形限制问题，解决方案的渐近行为仅由对热方程式的解决方案给出，而neumann边界条件为$ x = 0 $，初始datum $（2m/n）δ_{x = 0} $，其中m是我们原始问题的总质量，而我们的原始问题的总质量是$ n $ $ n $的数量，是$ n $的数量。此外，我们表明，当我们重新汇总内核时，非局部问题的解决方案会融合到热方程式（当地问题），也就是说，我们发现了一个松弛极限。
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 $(2M/N)δ_{x=0}$ 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.
