The evolution equation and the eigenvalue problem for the Laplacian in a regular tree

TL;DR

This paper investigates the evolution equation for the Laplacian on a regular tree, focusing on eigenvalues and long-term decay, providing existence, uniqueness, and asymptotic analysis of solutions.

Contribution

It introduces a detailed analysis of the eigenvalue problem for the Laplacian on regular trees, including existence, uniqueness, and decay rates of solutions.

Findings

Solutions exist and are unique under compatible initial conditions.

Solutions decay exponentially at a rate determined by the first eigenvalue.

Detailed analysis of the first eigenvalue for the Laplacian in regular trees.

Abstract

In this paper, our main goal is to study the evolution problem associated with the Laplacian operator with Dirichlet boundary conditions on a regular tree. To this end, we place special emphasis on the associated first eigenvalue problem, which provides the fundamental tool for describing the long-time dynamics. First, we prove existence and uniqueness of solutions when the initial condition is compatible with the boundary condition. Next, we address the asymptotic behavior of the solutions and show that they decay to zero exponentially fast. This decay rate is determined by the associated first eigenvalue, which we also analyze in detail.