Author:
Bhardwaj Arun Kumar,Kumar Vishvesh,Mondal Shyam Swarup
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 precisely, 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
$\mathcal {C}^1([0,\,T],\,H^1_{\mathcal {L}}(G)).$
Publisher
Cambridge University Press (CUP)