Asymptotic behavior for a class of viscoelastic equations with memory lacking instantaneous damping

Jiangwei Zhang; Yongqin Xie; Jiangwei Zhang; Yongqin Xie

doi:10.3934/math.2021552

AIMS Mathematics

2021, Volume 6, Issue 9: 9491-9509. doi: 10.3934/math.2021552

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Research article

Asymptotic behavior for a class of viscoelastic equations with memory lacking instantaneous damping

Jiangwei Zhang ,
Yongqin Xie ^,

School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, 410114, China

Received: 29 December 2020 Accepted: 20 June 2021 Published: 24 June 2021
MSC : 35K57, 35B40, 35B41

In this paper, we mainly investigate long-time behavior for viscoelastic equation with fading memory

$ u_{tt}-\Delta u_{tt}-\nu \Delta u+\int_{0}^{+\infty}k'(s)\Delta u(t-s)ds+f(u) = g(x). $

The main feature of the above equation is that the equation doesn't contain $ -\Delta u_t $, which contributes to a strong damping. The existence of global attractors is obtained by proving asymptotic compactness of the semigroup generated by the solutions for the viscoelastic equation. In addition, the upper semicontinuity of global attractors also is obtained.
- viscoelastic equation,
- contractive function,
- global attractor,
- memory
Citation: Jiangwei Zhang, Yongqin Xie. Asymptotic behavior for a class of viscoelastic equations with memory lacking instantaneous damping[J]. AIMS Mathematics, 2021, 6(9): 9491-9509. doi: 10.3934/math.2021552

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Abstract

In this paper, we mainly investigate long-time behavior for viscoelastic equation with fading memory

$ u_{tt}-\Delta u_{tt}-\nu \Delta u+\int_{0}^{+\infty}k'(s)\Delta u(t-s)ds+f(u) = g(x). $

The main feature of the above equation is that the equation doesn't contain $ -\Delta u_t $, which contributes to a strong damping. The existence of global attractors is obtained by proving asymptotic compactness of the semigroup generated by the solutions for the viscoelastic equation. In addition, the upper semicontinuity of global attractors also is obtained.

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