Estimates for the nonlinear viscoelastic damped wave equation on compact Lie groups
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| Format: | Preprint |
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2022
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| _version_ | 1866917671215300608 |
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| author | Bhardwaj, Arun Kumar Kumar, Vishvesh Mondal, Shyam Swarup |
| author_facet | Bhardwaj, Arun Kumar Kumar, Vishvesh Mondal, Shyam Swarup |
| contents | 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 $\mathcal{C}^1([0,T],H^1_{\mathcal L}(G)).$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2207_06645 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Estimates for the nonlinear viscoelastic damped wave equation on compact Lie groups Bhardwaj, Arun Kumar Kumar, Vishvesh Mondal, Shyam Swarup Analysis of PDEs Primary 35L15, 35L05, Secondary 35L05 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 $\mathcal{C}^1([0,T],H^1_{\mathcal L}(G)).$ |
| title | Estimates for the nonlinear viscoelastic damped wave equation on compact Lie groups |
| topic | Analysis of PDEs Primary 35L15, 35L05, Secondary 35L05 |
| url | https://arxiv.org/abs/2207.06645 |