Visco-elastic damped wave models with time-dependent coefficient
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917828326588416 |
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| author | Aslan, Halit Sevki Reissig, Michael |
| author_facet | Aslan, Halit Sevki Reissig, Michael |
| contents | In this paper, we study the following Cauchy problem for linear visco-elastic damped wave models with a general time-dependent coefficient $g=g(t)$: \begin{equation} \label{EqAbstract} \tag{$\star$} \begin{cases} u_{tt}- Δu + g(t)(-Δ)u_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} We are interested to study the influence of the damping term $g(t)(-Δ)u_t$ on qualitative properties of solutions to \eqref{EqAbstract} as decay estimates for energies of higher order and the parabolic effect. The main tools are related to WKB-analysis. We apply elliptic as well as hyperbolic WKB-analysis in different parts of the extended phase space. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_12340 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Visco-elastic damped wave models with time-dependent coefficient Aslan, Halit Sevki Reissig, Michael Analysis of PDEs 35L30, 35B40, 35L15, 35L05 In this paper, we study the following Cauchy problem for linear visco-elastic damped wave models with a general time-dependent coefficient $g=g(t)$: \begin{equation} \label{EqAbstract} \tag{$\star$} \begin{cases} u_{tt}- Δu + g(t)(-Δ)u_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} We are interested to study the influence of the damping term $g(t)(-Δ)u_t$ on qualitative properties of solutions to \eqref{EqAbstract} as decay estimates for energies of higher order and the parabolic effect. The main tools are related to WKB-analysis. We apply elliptic as well as hyperbolic WKB-analysis in different parts of the extended phase space. |
| title | Visco-elastic damped wave models with time-dependent coefficient |
| topic | Analysis of PDEs 35L30, 35B40, 35L15, 35L05 |
| url | https://arxiv.org/abs/2307.12340 |