Pullback attractors for a class of non-autonomous thermoelastic plate systems

Flank D. M. Bezerra; Vera L. Carbone; Marcelo J. D. Nascimento; Karina Schiabel

doi:10.3934/dcdsb.2017214

Article Contents

2018, Volume 23, Issue 9: 3553-3571. Doi: 10.3934/dcdsb.2017214

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Pullback attractors for a class of non-autonomous thermoelastic plate systems

1.
Universidade Federal da Paraíba Departamento de Matemática 58051-900 João Pessoa PB, Brazil
2.
Universidade Federal de São Carlos, Departamento de Matemática, 13565-905 São Carlos SP, Brazil

* Corresponding author
* Corresponding author

The first author is partially supported by FAPESP grant #2014/03686-3, Brazil.
The third author is partially supported by FAPESP grant #2014/03109-6, Brazil.

Received: March 31, 2017

Revised: June 30, 2017

Early access: August 2017

Published: September 2018

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Abstract

In this article we study the asymptotic behavior of solutions, in the sense of pullback attractors, of the evolution system

$\begin{cases}u_{tt} +Δ^2 u+a(t)Δθ=f(t,u),&t>τ,\ x∈Ω,\\θ_t-κΔ θ-a(t)Δ u_t=0,&t>τ,\ x∈Ω,\end{cases}$

subject to boundary conditions

$u=Δ u=θ=0,\ t>τ,\ x∈\partial Ω,$

where $Ω$ is a bounded domain in $\mathbb{R}^N$ with $N≥ 2$, which boundary $\partialΩ$ is assumed to be a $\mathcal{C}^4$-hypersurface, $κ>0$ is constant, $a$ is an Hölder continuous function and $f$ is a dissipative nonlinearity locally Lipschitz in the second variable. Using the theory of uniform sectorial operators, in the sense of P. Sobolevskiǐ ([23]), we give a partial description of the fractional power spaces scale for the thermoelastic plate operator and we show the local and global well-posedness of this non-autonomous problem. Furthermore we prove existence and uniform boundedness of pullback attractors.

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Mathematics Subject Classification: Primary: 37B55, 34D45; Secondary: 35B40, 35B41.

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References

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