### 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 language | English (US) |
---|---|

Pages (from-to) | 3496-3517 |

Number of pages | 22 |

Journal | Journal of Differential Equations |

Volume | 246 |

Issue number | 9 |

DOIs | |

State | Published - May 1 2009 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

*Journal of Differential Equations*,

*246*(9), 3496-3517. https://doi.org/10.1016/j.jde.2009.02.020

}

*Journal of Differential Equations*, vol. 246, no. 9, pp. 3496-3517. https://doi.org/10.1016/j.jde.2009.02.020

**Global attractors for the extensible thermoelastic beam system.** / Giorgi, C.; Naso, M. G.; Pata, V.; Potomkin, Mykhailo.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Global attractors for the extensible thermoelastic beam system

AU - Giorgi, C.

AU - Naso, M. G.

AU - Pata, V.

AU - Potomkin, Mykhailo

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.

UR - http://www.scopus.com/inward/record.url?scp=63149111993&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=63149111993&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2009.02.020

DO - 10.1016/j.jde.2009.02.020

M3 - Article

AN - SCOPUS:63149111993

VL - 246

SP - 3496

EP - 3517

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 9

ER -