Semilinear wave equation on compact Lie groups

TL;DR

This paper investigates the semilinear wave equation on compact Lie groups, establishing local existence, blow-up conditions, and lifespan estimates for solutions, advancing understanding of nonlinear wave behavior in geometric settings.

Contribution

It provides the first analysis of semilinear wave equations on compact Lie groups, including existence, blow-up, and lifespan results using Fourier analysis techniques.

Findings

Local existence in energy space proven

Blow-up occurs for all p>1 under certain initial conditions

Sharp lifespan estimates derived for solutions

Abstract

In this note, we study the semilinear wave equation with power nonlinearity $|u|^p$ on compact Lie groups. First, we prove a local in time existence result in the energy space via Fourier analysis on compact Lie groups. Then, we prove a blow-up result for the semilinear Cauchy problem for any $p>1$, under suitable sign assumptions for the initial data. Furthermore, sharp lifespan estimates for local (in time) solutions are derived.