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Optimality on Polynomial decay of semigroups in
Elasticity
By
Jaime E. Muñoz Rivera
LNCCIM-UFRJ
Our problem:
Given a semigroup of contractions T(t) defined over Hilbert space H, find the best number p such that
Liu and Rao proved a sufficient condition to get the polynomial decay of a semigroup
Removing logarits, Liu and Rao’s Theorem can be written as:
A sufficient and a necessary condition to polynomial decay was given by J. Pruss 2006
Other importante result due to Pruss 2006
Our purporse is to show that the sufficient condition of Liu and Rao is also a necessary condition
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.
The nexus between Liu-Rao and Pruss Characterization is given by a result due to
This is a join work with Luci Fatori:
Estadual University of LondrinaParaná – Brasil e-mail: [email protected]
Our interest is to prove that the sufficient condition of Liu and Rao is also a necessary condition.
This result will be important to show when a rate of decay is optimal.
Our main result is the following necessary condition
The proof is based on Pruss necessary condition and Latushkin –Shvidkoy result.
The Infenitesimal generator of the semigroup is
We denote the associated semigroup as
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.
Z. Liu and K. Liu, proved that the semigroup is analytic when
and Differentiable when
Our contribution is about polynomial stability for
Our stability result to damped wave equation is
Idea of the proof of the optimality:
