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- Title
Estimates for the nonlinear viscoelastic damped wave equation on compact Lie groups.
- Authors
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 $\mathcal {C}^1([0,\,T],\,H^1_{\mathcal {L}}(G)).$
- Subjects
COMPACT groups; WAVE equation; NONLINEAR wave equations; LIE groups; CAUCHY problem; NONLINEAR equations
- Publication
Proceedings of the Royal Society of Edinburgh: Section A: Mathematics, 2024, Vol 154, Issue 3, p810
- ISSN
0308-2105
- Publication type
Article
- DOI
10.1017/prm.2023.38