Uniqueness class of solutions to a class of linear evolution equations

TL;DR

This paper investigates the uniqueness of solutions to wave equations on infinite graphs, demonstrating non-uniqueness in some cases and establishing sharp conditions for uniqueness, with extensions to other linear evolution equations.

Contribution

It constructs examples of non-uniqueness and derives sharp uniqueness classes for solutions, extending results to a broad class of linear evolution equations.

Findings

Constructed non-uniqueness examples for wave equations on graphs.

Established sharp conditions for solution uniqueness.

Extended results to various linear evolution equations.

Abstract

In this paper, we study the wave equation on infinite graphs. On one hand, in contrast to the wave equation on manifolds, we construct an example for the non-uniqueness for the Cauchy problem of the wave equation on graphs. On the other hand, we obtain a sharp uniqueness class for the solutions of the wave equation. The result follows from the time analyticity of the solutions to the wave equation in the uniqueness class. In the last part, we extend the result to a wide class of linear evolution equations.