A nonlinear Klein-Gordon equation on a star graph

TL;DR

This paper investigates the well-posedness and stability of solutions to a nonlinear Klein-Gordon equation on a star graph, employing fixed point methods and linearization techniques to extend stability results from related NLS models.

Contribution

It establishes local well-posedness and analyzes the orbital stability and instability of standing waves for the nonlinear Klein-Gordon equation on a star graph, introducing new analytical approaches.

Findings

Proves local well-posedness using fixed point and Hille-Yosida theorem.

Analyzes stability and instability of standing waves.

Extends stability results from NLS equations with delta-interaction.

Abstract

We study local well-posedness and orbital stability/instability of standing waves for a first order system associated with a nonlinear Klein-Gordon equation on a star graph. The proof of the well-posedness uses a classical fixed point argument and the Hille-Yosida theorem. Stability study relies on the linearization approach and recent results for the NLS equation with the $\delta$-interaction on a star graph.