On the extensible viscoelastic beam

This work is focused on the equation $$ \ptt u+\partial_{xxxx}u +\int_0^\infty \mu(s) \partial_{xxxx}[u(t)-u(t-s)]ds- \big(\beta+\|\partial_x u\|_{L^2(0,1)}^2\big)\partial_{xx}u= f$$ describing the motion of an extensible viscoelastic beam. Under suitable boundary conditions, 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 appropriate decomposition of the solution semigroup, together with the existence of a Lyapunov functional.