Regular global attractors of type III thermoelastic extensible beams

Abstract

For β ∈ R, the authors consider the evolution system in the unknown variables u and α ∂ttu + ∂xxxxudescribing the dynamics of type III thermoelastic extensible beams, where the dissipation is entirely contributed by the second equation ruling the evolution of the thermal displacement α. Under natural boundary conditions, the existence of the global attractor of optimal regularity for the related dynamical system acting on the phase space of weak energy solutions is established.

Some decay properties in extensible thermoelastic beam with Gurtin–Pipkin’s law

“…This is one of the reasons that has motivated some authors to develop the extensible elastic beam theory in thermal environment. Some papers dealt with extensible thermoelastic beams (see, e.g., [1][2][3][4][5][6][7][8][9], among others). In most of these papers, one employs Fourier's law for the physical derivation of thermal effects on the mechanical behaviors of beams…”

“…Also Potomkin [9] studied the long-time behavior of two-dimensional extensible thermoviscoelastic plate equations due to Berger approach under Coleman–Gurtin’s and Gurtin–Pipkin’s laws. Some properties of solutions to thermoelastic extensible beams have been studied by Conti et al [12, 13], Coti Zelati [5], Dell’Oro and Pata [6], where the conduction law proposed by Green and Naghdi has been proposed instead of Fourier’s law. Recently, Aouadi [1] derived the equations that constitute the nonlinear mathematical model of extensible elastic beam with temperature and microtemperature effects.…”