Global attractors for the extensible thermoelastic beam system

Abstract

This work is focused on the dissipative system { ∂ t t u + ∂ x x x x u + ∂ x x θ − ( β + ‖ ∂ x u ‖ L 2 ( 0 , 1 ) 2 ) ∂ x x u = f , ∂ t θ − ∂ x x θ − ∂ x x t u = g describing the dynamics of an extensible thermoelastic beam, where the dissipation is entirely contributed by the second equation ruling the evolution of θ. Under natural boundary conditions, we prove the existence of bounded absorbing sets. When the external sources f and g are time-independent, the related semigroup of solutions is shown to possess the global attractor of optimal regularity for all parameters β ∈ R . The same result holds true when the first equation is replaced by ∂ t t u − γ ∂ x x t t u + ∂ x x x x u + ∂ x x θ − ( β + ‖ ∂ x u ‖ L 2 ( 0 , 1 ) 2 ) ∂ x x u = f with γ > 0 . In both cases, the solutions on the attractor are strong solutions.