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ON THE EXTENSIBLE VISCOELASTIC BEAM

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IOP PUBLISHING NONLINEARITY

Nonlinearity 21 (2008), 713-733 doi: 10.1088/0951-7715/21/004 Received 19 January 2008

Published 28 February 2008

Online at stacks.iop.org/Non/21/713 Recommended by E.S. Titi

ON THE EXTENSIBLE VISCOELASTIC BEAM

CLAUDIO GIORGI, VITTORINO PATA, ELENA VUK

Abstract. This work is focused on the equation

ttu + ∂xxxxu + Z

0

µ(s)∂xxxx[u(t) − u(t − s)]ds − β + k∂xuk2L2(0,1)∂xxu = f describing the motion of an extensible viscoelastic beam. Under suitable boundary con- ditions, the related dynamical system in the history space framework is shown to possess a global attractor of optimal regularity. The result is obtained by exploiting an appropri- ate decomposition of the solution semigroup, together with the existence of a Lyapunov functional.

Dipartimento di Matematica, Universit`a di Brescia Via Valotti 9, 25133 Brescia, Italy

E-mail address: giorgi@ing.unibs.it E-mail address: vuk@ing.unibs.it

Dipartimento di Matematica “F.Brioschi”, Politecnico di Milano Via Bonardi 9, 20133 Milano, Italy

E-mail address: vittorino.pata@polimi.it

2000 Mathematics Subject Classification. 35B41,35B65,37B25,45K05,74K10.

Key words and phrases. Viscoelastic beam equation, memory effects, Lyapunov functional, regular global attractors.

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