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Optimality on Polynomial decay of semigroups in Elasticity By Jaime E. Muñoz Rivera LNCC IM-UFRJ

Optimality on Polynomial decay of semigroups in Elasticity By Jaime E. Muñoz Rivera LNCC IM-UFRJ. Our problem :. Given a semigroup of contractions T(t) defined over Hilbert space H, find the best number p such that. know result about Polynomial stability.

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Optimality on Polynomial decay of semigroups in Elasticity By Jaime E. Muñoz Rivera LNCC IM-UFRJ

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  1. Optimality on Polynomial decay of semigroups in Elasticity By Jaime E. Muñoz Rivera LNCC IM-UFRJ

  2. Our problem: Given a semigroup of contractions T(t) defined over Hilbert space H, find the best number p such that

  3. knowresultabout Polynomialstability

  4. Liu and Rao proved a sufficient condition to get the polynomial decay of a semigroup

  5. Removing logarits, Liu and Rao’s Theorem can be written as:

  6. A sufficient and a necessary condition to polynomial decay was given by J. Pruss 2006

  7. Other importante result due to Pruss 2006

  8. Pruss method give a necessary and a sufficient condition to prove polynomial estability The problem is that it is not a simple task to estimate fractional powers of the operator of I.G.S. It is more easy to deal with the sufficient condition given by Liu and Rao. Our purporse is to show that the sufficient condition of Liu and Rao is also a necessary condition

  9. The nexus between Liu-Rao and Pruss Characterization is given by a result due to

  10. This is a join work with Luci Fatori: Estadual University of Londrina Paraná – Brasil e-mail: lucifatori@uel.br Our interest is to prove that the sufficient condition of Liu and Rao is also a necessary condition.

  11. Our main result is the following necessary condition This result will be important to show when a rate of decay is optimal. The proof is based on Pruss necessary condition and Latushkin –Shvidkoy result.

  12. Aplications

  13. The Infenitesimal generator of the semigroup is We denote the associated semigroup as

  14. Previousresult

  15. That system was studied by Chen and Triggiani. They proved that the semigroup is analytic if The authors solved the conjetures of G. Chen and D. L. Russel on structural damping for elastic systems.

  16. Z. Liu and K. Liu, proved that the semigroup is analytic when and Differentiable when

  17. Our Contribution

  18. Our contribution is about polynomial stability for Our stability result to damped wave equation is

  19. Idea oftheproofoftheoptimality:

  20. Our contribution to Bresse system

  21. Thanks

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