Estimates for the nonlinear viscoelastic damped wave equation on compact Lie groups

TL;DR

This paper studies the behavior of solutions to a nonlinear viscoelastic damped wave equation on compact Lie groups, providing $L^2$ estimates, decay rate limitations, and local existence results using noncommutative Fourier analysis.

Contribution

It introduces new $L^2$-estimates for solutions, demonstrates decay rate limitations under certain conditions, and establishes local existence in the energy space on compact Lie groups.

Findings

No decay rate improvement with additional $L^1(G)$-regularity.

Established $L^2$-estimates for solutions.

Proved local existence in energy space using noncommutative Fourier analysis.

Abstract

Let $G$ be a compact Lie group. In this article, we investigate the Cauchy problem for a nonlinear wave equation with the viscoelastic damping on $G$. More preciously, we investigate some $L^2$-estimates for the solution to the homogeneous nonlinear viscoelastic damped wave equation on $G$ utilizing the group Fourier transform on $G$. We also prove that there is no improvement of any decay rate for the norm $\|u(t,\cdot)\|_{L^2(G)}$ by further assuming the $L^1(G)$-regularity of initial data. Finally, using the noncommutative Fourier analysis on compact Lie groups, we prove a local in time existence result in the energy space $\mathcal{C}^1([0,T],H^1_{\mathcal L}(G)).$