Nonlinear fractional wave equation on compact Lie groups

TL;DR

This paper studies fractional wave equations on compact Lie groups, establishing estimates, local existence, blow-up conditions, lifespan, and well-posedness for related Klein-Gordon equations using Fourier analysis.

Contribution

It introduces new $L^{2}$ estimates and proves local existence, blow-up, and lifespan results for fractional wave equations on compact Lie groups, extending analysis to Klein-Gordon equations.

Findings

Established $L^{2}$ estimates for solutions.

Proved local in-time existence in energy space.

Derived conditions for finite time blow-up and sharp lifespan.

Abstract

Let $G$ be a compact Lie group. In this article, we consider the initial value fractional wave equation with power-type nonlinearity on $G$. Mainly, we investigate some $L^{2}-L^{2}$ estimates of the solutions to the homogeneous fractional wave equation on $G$ with the help of the group Fourier transform on $G$. Further, using the Fourier analysis on compact Lie groups, we prove a local in-time existence result in the energy space. Moreover, under certain conditions on the initial data, a finite time blow-up result is established. We also derive a sharp lifespan for local (in-time) solutions. Finally, we consider the space-fractional wave equation with a regular mass term depending on the position and study the well-posedness of the fractional Klein-Gordon equation on compact Lie groups.