Global attractors for the extensible thermoelastic beam system

C. Giorgi, M. G. Naso, V. Pata, M. Potomkin

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Abstract

This work is focused on the dissipative system{(∂t t u + ∂x x x x u + ∂x x θ - (β + {norm of matrix} ∂x u {norm of matrix}L2 (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 + {norm of matrix} ∂x u {norm of matrix}L2 (0, 1)2) ∂x x u = f with γ > 0. In both cases, the solutions on the attractor are strong solutions.

Original languageEnglish (US)
Pages (from-to)3496-3517
Number of pages22
JournalJournal of Differential Equations
Volume246
Issue number9
DOIs
StatePublished - May 1 2009

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All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

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