TY - JOUR
T1 - Global attractors for the extensible thermoelastic beam system
AU - Giorgi, C.
AU - Naso, M. G.
AU - Pata, V.
AU - Potomkin, M.
PY - 2009/5/1
Y1 - 2009/5/1
N2 - 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.
AB - 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.
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U2 - 10.1016/j.jde.2009.02.020
DO - 10.1016/j.jde.2009.02.020
M3 - Article
AN - SCOPUS:63149111993
SN - 0022-0396
VL - 246
SP - 3496
EP - 3517
JO - Journal of Differential Equations
JF - Journal of Differential Equations
IS - 9
ER -