Blow-up solutions of damped Klein-Gordon equation on the Heisenberg group

TL;DR

This paper proves that solutions to a damped Klein-Gordon equation on the Heisenberg group blow up in finite time for any positive initial energy, extending previous work on global well-posedness for small energies.

Contribution

It establishes finite-time blow-up for solutions with arbitrary positive initial energy on the Heisenberg group, complementing prior results on small energy solutions.

Findings

Solutions blow up in finite time for positive initial energy.

Blow-up occurs regardless of initial energy size.

Extends understanding of Klein-Gordon dynamics on the Heisenberg group.

Abstract

Inthisnote,weprovetheblow-upofsolutionsofthesemilineardamped Klein-Gordon equation in a finite time for arbitrary positive initial energy on the Heisenberg group. This work complements the paper [21] by the first author and Tokmagambetov, where the global in time well-posedness was proved for the small energy solutions.