Evolution models with time-dependent coefficients in friction and viscoelastic damping terms
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arXiv
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| Format: | Preprint |
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2025
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| author | Aslan, Halit Sevki Reissig, Michael |
| author_facet | Aslan, Halit Sevki Reissig, Michael |
| contents | We study the following Cauchy problem for the linear wave equation with both time-dependent friction and time-dependent viscoelastic damping: \begin{equation} \label{EqAbstract}\tag{$\ast$} \begin{cases} u_{tt}- Δu + b(t)u_t - g(t)Δu_t=0, &(t,x) \in (0,\infty) \times \mathbb{R}^n, \\ u(0,x)= u_0(x),\quad u_t(0,x)= u_1(x), &x \in \mathbb{R}^n. \end{cases} \end{equation} Our goal is to derive decay estimates for higher order energy norms of solutions to this problem. We focus on the interplay between the time-dependent coefficients in both damping terms and their influence on the qualitative behavior of solutions. The analysis is based on a classification of the damping mechanisms, frictional damping $b(t)u_t$ and viscoelastic damping $-g(t)Δu_t$ as well, and employs the WKB-method in the extended phase space. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_24058 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Evolution models with time-dependent coefficients in friction and viscoelastic damping terms Aslan, Halit Sevki Reissig, Michael Analysis of PDEs 35L05, 35L15, 35B40 We study the following Cauchy problem for the linear wave equation with both time-dependent friction and time-dependent viscoelastic damping: \begin{equation} \label{EqAbstract}\tag{$\ast$} \begin{cases} u_{tt}- Δu + b(t)u_t - g(t)Δu_t=0, &(t,x) \in (0,\infty) \times \mathbb{R}^n, \\ u(0,x)= u_0(x),\quad u_t(0,x)= u_1(x), &x \in \mathbb{R}^n. \end{cases} \end{equation} Our goal is to derive decay estimates for higher order energy norms of solutions to this problem. We focus on the interplay between the time-dependent coefficients in both damping terms and their influence on the qualitative behavior of solutions. The analysis is based on a classification of the damping mechanisms, frictional damping $b(t)u_t$ and viscoelastic damping $-g(t)Δu_t$ as well, and employs the WKB-method in the extended phase space. |
| title | Evolution models with time-dependent coefficients in friction and viscoelastic damping terms |
| topic | Analysis of PDEs 35L05, 35L15, 35B40 |
| url | https://arxiv.org/abs/2506.24058 |