Nonlinear fractional damped wave equation on compact Lie groups

TL;DR

This paper investigates the fractional damped wave equation on compact Lie groups, establishing local existence, blow-up conditions, and global solutions for small data using Fourier analysis.

Contribution

It introduces new results on local and global existence, as well as blow-up phenomena, for nonlinear fractional wave equations on compact Lie groups.

Findings

Proved local in-time existence in the energy space.

Established finite time blow-up under certain initial conditions.

Proved global existence for small data solutions with damping and mass.

Abstract

In this paper, we deal with the initial value fractional damped wave equation on $G$, a compact Lie group, with power-type nonlinearity. The aim of this manuscript is twofold. First, using the Fourier analysis on compact Lie groups, we prove a local in-time existence result in the energy space for the fractional damped wave equation on $G$. Moreover, a finite time blow-up result is established under certain conditions on the initial data. In the next part of the paper, we consider fractional wave equation with lower order terms, that is, damping and mass with the same power type nonlinearity on compact Lie groups, and prove the global in-time existence of small data solutions in the energy evolution space.