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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.06414 |
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| _version_ | 1866908756930985984 |
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| author | Liu, Linfang Narciso, Vando Yang, Zhijian |
| author_facet | Liu, Linfang Narciso, Vando Yang, Zhijian |
| contents | This article aims to study the long-time dynamics of the linear viscoelastic plate equation $\displaystyle{u_{tt}+Δ^2 u-\int_τ^tg(t-s)Δ^2u(s)ds=0}$ subject to nonlinear and nonlocal boundary conditions. This model, with $τ=0$, was first considered by Cavalcanti (Discrete Contin. Dyn. Syst., 8(3), 675-695, 2002), where results of global existence and uniform decay rates of energy have been established. In this work, by taking $τ=-\infty$, and considering the autonomous equivalent problem we prove that the dynamical system $(\mathcal{H},S_t)$ generated by the weak solutions has a compact global attractor $\mathfrak{A}$ (in the topology of the weak phase space $\mathcal{H}$), which in subcritical case has finite dimension and smoothness. Furthermore, when the force follows the {\it Hook Law}, we prove that $(\mathcal{H},S_t)$ possesses a (generalized) fractal exponential attractor $\mathfrak{A}_{\exp}$ with finite dimension in a space $\widetilde{\mathcal{H}}\supset\mathcal{H}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_06414 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Dynamics for a viscoelastic beam equation with past history and nonlocal boundary dissipation Liu, Linfang Narciso, Vando Yang, Zhijian Analysis of PDEs This article aims to study the long-time dynamics of the linear viscoelastic plate equation $\displaystyle{u_{tt}+Δ^2 u-\int_τ^tg(t-s)Δ^2u(s)ds=0}$ subject to nonlinear and nonlocal boundary conditions. This model, with $τ=0$, was first considered by Cavalcanti (Discrete Contin. Dyn. Syst., 8(3), 675-695, 2002), where results of global existence and uniform decay rates of energy have been established. In this work, by taking $τ=-\infty$, and considering the autonomous equivalent problem we prove that the dynamical system $(\mathcal{H},S_t)$ generated by the weak solutions has a compact global attractor $\mathfrak{A}$ (in the topology of the weak phase space $\mathcal{H}$), which in subcritical case has finite dimension and smoothness. Furthermore, when the force follows the {\it Hook Law}, we prove that $(\mathcal{H},S_t)$ possesses a (generalized) fractal exponential attractor $\mathfrak{A}_{\exp}$ with finite dimension in a space $\widetilde{\mathcal{H}}\supset\mathcal{H}$. |
| title | Dynamics for a viscoelastic beam equation with past history and nonlocal boundary dissipation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2601.06414 |