Upper semicontinuity of optimal attractors for viscoelastic equations lacking strong damping

ABSTRACT

This paper devotes to investigating the asymptotic behavior of viscoelastic equation with fading memory lacking strong damping term $u_{tt}-\mathrm{\Delta }{u}_{tt}-\nu \mathrm{\Delta }u+{\int }_0^{+\mathrm{\infty }}\mu (s)\mathrm{\Delta }u(t-s)\text{d}s+\lambda u_t+f(u)=g(x),$where $\lambda \in [0,1)$. Its key feature is the lack of strong mechanical damping term $-\mathrm{\Delta }{u}_t$ replaced by weaker dissipative term (i.e. memory term). At present, the existing results only considered the existence and regularity of global attractors, but do not obtain the existence of the strong attractors, because the compactness is difficult to get in highly regular space. Accordingly, in current work, we will prove the existence of strongly global attractors with only weak dissipation when the nonlinearity f satisfies critical growth by using the dominated convergence theorem and the asymptotically contractive function method. Moreover, the upper semicontinuity of the strongly global attractors is also obtained when the perturbed parameter $\lambda \to 0$, which deepens some conclusions that exist in Conti et al. [Asymptotics of viscoelastic materials with nonlinear density and memory effects. J Differ Equ. 2018;264:4235–4259] and Qin et al. [Uniform attractors for a nonautonomous viscoelastic equation with a past history. Nonlinear Theory Methods Appl. 2014;101:1–15].

KEYWORDS:

• Viscoelastic equation
• critical growth
• strongly global attractor
• upper semicontinuity
• memory

AMS SUBJECT CLASSIFICATIONS::

• 35K57
• 35B40
• 35B41

Acknowledgments

The authors would like to thank the referees for their many helpful comments and suggestions.

Disclosure statement

The authors have no conflicts to disclose.

Additional information

Funding

The research is supported by National Natural Science Foundation of China [No. 11771449] and Natural Science Foundation of Hunan [No. 2020JJ4102].