Regularity and upper semicontinuity of pullback attractors for a class of nonautonomous thermoelastic plate systems

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

Vol. 301 (2019), No. 2, 395–419

DOI: 10.2140/pjm.2019.301.395

Abstract

We study the long-time dynamics, in the sense of pullback attractors, of solutions for semilinear nonautonomous thermoelastic plate systems in a bounded smooth domain in ${ℝ}^N$, $N\ge 2$. Using the theory of uniform sectorial operators, in the sense of P. Sobolevskiĭ (1961), we will prove existence, uniform boundedness, regularity and upper semicontinuity of pullback attractors for the evolution system

$$\left\{\begin{array}{cc}{u}_{tt}+{\Delta }^{2}u+a\Delta \theta =f\left(u\right),\hfill & t>\tau ,x\in \Omega ,\hfill \\ {\theta }_t-\kappa \left(t\right)\Delta \theta -a\Delta u_t=0,\hfill & t>\tau ,x\in \Omega ,\hfill \end{array}\right.$$

subject to boundary conditions

$$u=\Delta u=\theta =0,t>\tau ,x\in \partial \Omega ,$$

with respect to the functional parameter $\kappa$.

Keywords

pullback attractors, regularity, thermoelastic plate systems, upper semicontinuity

Mathematical Subject Classification 2010

Primary: 35B40, 35B41, 35B65, 37B55

Milestones

Received: 27 September 2018

Revised: 12 January 2019

Accepted: 17 January 2019

Published: 24 October 2019