On the blow-up of solutions to semilinear damped wave equations with power nonlinearity in compact Lie groups

TL;DR

This paper proves that solutions to certain semilinear damped wave equations on compact Lie groups blow up in finite time for any power nonlinearity greater than one, providing lifespan estimates and local existence results.

Contribution

It establishes blow-up results and lifespan bounds for semilinear damped wave equations on compact Lie groups, extending understanding to this geometric setting.

Findings

Solutions blow up in finite time for all p>1

Provides upper bounds for the lifespan of solutions

Establishes local existence in the energy space

Abstract

In this note, we prove a blow-up result for the semilinear damped wave equation in a compact Lie group with power nonlinearity $|u|^p$ for any $p>1$, under suitable integral sign assumptions for the initial data, by using an iteration argument. A byproduct of this method is the upper bound estimate for the lifespan of a local in time solution. As a preliminary result, a local (in time) existence result is proved in the energy space via Fourier analysis on compact Lie groups.